Short Answer:
A relatively minor change in a planet's mass can cause larger changes in its surface gravity, escape velocity, and abilities to produce and maintain an atmosphere.
Examples exist in our solar system of various planets and moons with atmospheres, and there are cases where the relative densities of the atmospheres of various worlds do not correspond to their relative abilities to produce or retain an atmosphere.
Thus while most planets with surface gravities of 3 g would be expected to have atmospheres much denser than that of Earth, it is possible that some planets with such high surface gravity have atmospheres much thinner than that of Earth making it impossible to fly there.
While most planets with surface gravity lower than that of Earth might have atmospheres less thick than Earth's it is possible that some planets with surface gravity lower than that of Earth could have atmospheres more dense that of earth, and thus be easier to flying, if flying machines or living flying creatures can function in their environments. In fact two such worlds are known to exist in our solar system and would be much better places to fly that Earth is - for animals who can survive in or machines designed to operate in their environments.
Thus you might consider making your fictional planet one with lower gravity than Earth but a much denser atmosphere than Earth's atmosphere, thus making it a much better place to fly.
If the point of your story idea is a planet with a surface gravity of 3 g, which would thus be much harder to fly in than Earth, but which has a much denser atmosphere than Earth's, so making it surprisingly easy to fly in, then you should go ahead with a planet with a surface gravity of 3 g, but you should be aware of various factors that a planet with a higher gravity and a denser atmosphere than Earths would have.
1) Humans might not be able to breath and survive in an atmosphere dense enough for your story needs. Every possible atmospheric gas will be toxic to humans at a high enough pressure. Even oxygen, necessary to survive, can kill humans at a high enough pressure. So if you add up the highest survivable atmospheric pressure of every possible gas, you will come to an absolute maximum possible survivable atmospheric pressure for humans. If an even higher atmospheric pressure is needed for flight on your planet, any possible human visitors will have to use environmental suits, and at an even higher atmospheric pressure it is possible that no native lifeforms could survive and the planet would be lifeless.
2) Humans could not survive more than about 1.5 g for long, so any possible human visitors to a 3 g planet would have to make very short visits and/or use some form of anti gravity.
3) a planet large enough to have a surface gravity of 3 g might possibly have to be totally covered with oceans, which would have various implications for the possibility of life on the planet and for the type of story one could write.
Long Answer:
The density of a planet's atmosphere depends on two factors:
1) The rate at which it produces and acquires its atmosphere via several different processes.
2) The rate at which it loses its atmosphere via several different processes.
If 2) is greater than 1) the planet will be losing its atmosphere over time and the atmosphere's pressure will be decreasing. The greater the difference, the more rapid the loss.
If 1) is greater than 2) the planet will be gaining atmosphere over time and the atmosphere's presser will be increasing. The greater the difference, the more rapid the gain.
Since 1) and 2) are both the results of several different processes, some of which can vary in intensity over time, the rate at which a planetary atmosphere increases or decreases can vary over the history the planet.
The ability of a planet to retain an atmosphere depends mainly on its escape velocity (formula at https://en.wikipedia.org/wiki/Escape_velocity1), not on its surface gravity (formula at https://en.wikipedia.org/wiki/Surface_gravity2).
The two formulas are not identical, so the differences between the surface gravities and the escape velocities of two different planets will not be in the same ratio. Since a lower surface gravity makes flying easier, and a higher escape velocity makes retaining a denser atmosphere more probable, science fiction writers should learn to exploit the differences between surface gravity and escape velocity.
Both the surface gravity and the escape velocity of a planet decrease slightly with altitude above the planet's surface. Flying will be easiest where the atmosphere is thickest closer to the surface, while molecules, atoms, and ions will escape from the highest and thinnest layers of the atmosphere, tens, hundreds, or thousands of kilometers or miles above the surface.
If your fictional planet has a surface gravity of 3g, human explorers could not survive on its surface long unless they have anti gravity belts to reduced the effect of the gravity on them. Thus it will not be important whether humans explorers will be able to breath the denser air of that planet or have to wear spacesuits (possibly with anti gravity belts) when on its surface, since they definitely won't be spending much time there.
Thus the atmosphere can be many times more dense than would be breathable for humans, as long as it remains breathable for the native life forms.
Stephen H. Dole, in Habitable Planets for Man (1964, 2007) made a lot of estimates about what was necessary for a planet to be habitable for humans (which is a subset of being habitable for life forms in general; humans would die more or less instantly if teleported to some environments on Earth that are teaming with life).
On page 53 he begins the discussion of the range range for a planet habitable for humans.
On page 53 Dole said that since a surface gravity of about 1.5 g seemed like the maximum that humans would tolerate, and that corresponded to a planet with a mass of 2.35 earth masses, a radius of 1.25 Earth radii, and an escape velocity of 15.3 kilometers per second.
Since your question is about a planet with a surface gravity of 3g, I guess you probably never intended for your planet to be habitable for humans. If you did intend for your planet to be habitable for humans with a surface gravity of 3g, it should be back to the drawing board for your worldbuilding.
The minimum mass for a habitable planet would be the minimum mass necessary to have a n escape velocity high enough relative to the average velocity of air particles to retain an atmosphere for billions of years.
On page 54 Dole calculated the minimum size of a planet that could retain a breathable atmosphere for billions of years as 0.195 Earth's mass, with of 0.63 of Earth's radius and a surface gravity of 0.49 g. But Dole believed such a planet would be unable to produce an atmosphere dense enough to be breathable.
...To prevent atomic oxygen from escaping from the upper layers of its atmosphere, the planet's escape velocity must be of the order of five times the root-mean-square velocity of the oxygen atoms in the atmosphere. This is shown in figure 12 (see page 37)...then the escape velocity of the smallest planet capable of retaining atomic oxygen may be as low as 6.25 kilometers per second (5 X 1.25). Going back to figure 9, this may be seen to correspond to a planet having a mass of 0.195 Earth mass, a radius of 0.63 Earth radius, and a surface gravity of 0.49 g. Under the above assumptions, such a planet could theoretically hold an oxygen-rich atmosphere, but it would probably be much too small to produce one, as will be seen below.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf3
Note that this hypothetical planet would have an escape velocity of 6.25 kilometers per second, which is 0.5587 of Earth's escape velocity of 11.186 kilometers per second, but a surface gravity of of 0.49 g, which is 0.49 of Earth's surface gravity of 1.000 g. This is an example of a difference in planetary mass resulting differences in escape velocity and surface gravity that are not in the same ratio.
Dole calculated via various lines of reasoning two figures for the minimum mass necessary to produce a breathable atmosphere, 0.253 Earth mass, which he believed too low, and 0.57 Earth Mass, which he believed too high:
With 0.25 being too low, and 0.57 being too high, the appropriate value of mass for the smallest habitable planet must lie between those figures, somewhere in the vicinity of 0.4 Earth mass.
...This corresponds to a planet having a radius of 0.78 Earth radius and a surface gravity of 0.68 g.
https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf3
So the minimum mass necessary to produce a breathable for humans oxygen rich atmosphere would 0.4 Earth masses, corresponding to a radius of 0.78 Earth radius and a surface gravity of 0.68 g. But that is an estimated minimum mass between 0.25 and 0.57 Earth masses which are considered too low and too high respectively. It is always possible for the actual minimum mass to be higher, closer to 0.57 Earth masses, or lower, closer to 0.25 Earth masses.
Of course a lot has been learned about planetary atmospheres since Dole wrote in the early 1960s.
The planet Venus has a surface gravity of 0.904 g and an escape velocity 10.36 kilometers per second, which is 0.926 of earth's escape velocity of 11.186 kilometers per second. This is another example of the surface gravity and the escape velocity changing at different ratios as mass changes.
The ability of Venus to retain its atmosphere should be slightly less than that of Earth, and if a planet's ability to produce an atmosphere is directly proportional to its mass, Venus should have an atmosphere slightly less dense than that of Earth.
The atmosphere of Venus is the layer of gases surrounding Venus. It is composed primarily of carbon dioxide and is much denser and hotter than that of Earth. The temperature at the surface is 740 K (467 °C, 872 °F), and the pressure is 93 bar (9.3 MPa), roughly the pressure found 900 m (3,000 ft) underwater on Earth.1 The Venusian atmosphere supports opaque clouds made of sulfuric acid, making optical Earth-based and orbital observation of the surface impossible. Information about the topography has been obtained exclusively by radar imaging.1 Aside from carbon dioxide, the other main component is nitrogen. Other chemical compounds are present only in trace amounts.1
https://en.wikipedia.org/wiki/Atmosphere_of_Venus4
Any hypothetical alien life forms with alien biochemistry capable of surviving the temperatures on Venus would find flying in the dense atmosphere many times easier than flying on Earth.
Planetary scientists have done a lot of research and computer simulations over the last fifty years trying to explain the differences between the atmospheres of Venus and Earth.
Perhaps the most unexpected shock in the age of space probes exploring the Solar System was the atmosphere of Titan, the largest moon of Saturn. Although science fiction stories often depicted the larger moons of the giant planets with breathable atmospheres, astronomers believed they were all airless until traces of a thin methane atmosphere was were detected on Titan in 1948.
Here is a list of the largest terrestrial type planets and other bodies in the solar system, sorted by their escape velocities. Note that the ability of a body to retain an atmosphere for long periods of time also depends on it atmospheric temperature and thus in its distance from the Sun.
1) Pluto, the dwarf planet, has a mass 0.00218 that of Earth, a surface gravity of 0.063 g, and an escape velocity of 1.212 kilometers per second, 0.1083 that of Earth.
2) Triton, the moon of Neptune, has a mass 0.00359 that of Earth, a surface gravity of 0.0794 g, and an escape velocity of 1.455 kilometers per second, 0.13007 that of Earth.
3) Europa, a moon of Jupiter, has a mass 0.008 that of Earth, a surface gravity of 0.134 g, and an escape velocity of 2.025 kilometers per second, 0.1810 that of Earth.
4) The Moon, the moon of Earth, has a mass 0.012300 that of Earth, a surface gravity of 0.1654 g, and an escape velocity of 2.38 kilometers per second, 0.2127659 that of Earth.
5) Callisto, a moon of Jupiter, has a mass 0.018 that of Earth, a surface gravity of 0.126 g, and an escape velocity of 2.440 kilometers per second, 0.2181 that of Earth.
6) Io, a moon of Jupiter, has a mass 0.015 that of Earth, a surface gravity of 0.183 g, and an escape velocity of 2.588 kilometers per second, 0.2286 that of Earth.
7) Titan, a moon of Saturn, has a mass 0.0225 that of Earth, a surface gravity of 0.138 g, and an escape velocity of 2.639 kilometers per second, 0.2359 that of Earth.
8) Ganymede, a moon of Jupiter, has a mass 0.025 that of Earth, a surface gravity of 0.146 g, and an escape velocity of 2.741 kilometers per second, 0.2450 that of Earth.
9) The Planet Mercury, has a mass 0.055 that of Earth, a surface gravity of 0.38 g, and an escape velocity of 4.25 kilometers per second, 0.3799 that of Earth.
10) The Planet Mars, has a mass 0.107 that of Earth, a surface gravity of 0.3794 g, and an escape velocity of 5.027 kilometers per second, 0.4494 that of Earth.
Dole calculated that a planet with a moss of 0.195 that of Earth, a surface gravity of 0.49 g, and an escape velocity of 6.25 kilometers per second, 0.5587 that of Earth, could retain an oxygen rich atmosphere.
Dole estimated that a planet with the minimum mass necessary to produce an oxygen rich atmosphere would have 0.4 times the mass of earth, and a surface gravity of 0.68 g.
11) The Planet Venus, has a mass 0.815 that of Earth, a surface gravity of 0.904 g, and an escape velocity of 10.36 kilometers per second, 0.926 that of Earth.
12) The Planet Earth, has a mass 1.000 that of Earth, a surface gravity of 1 g, and an escape velocity of 11.186 kilometers per second, 1.000 that of Earth.
Putting them in the order of their atmospheric density:
1) The Moon, the moon of Earth, has a mass 0.012300 that of Earth, a surface gravity of 0.1654 g, an escape velocity of 2.38 kilometers per second, 0.2127659 that of Earth, and a surface pressure about 0.000000000000003 that of Earth.
2) The Planet Mercury, has a mass 0.055 that of Earth, a surface gravity of 0.38 g, and an escape velocity of 4.25 kilometers per second, 0.3799 that of Earth, and an atmospheric pressure of about 1 nanopascal or 0.000000001 Pascal, approximately one hundred trillionth (0.000000000000001) that of Earth.
3) Europa, a moon of Jupiter, has a mass 0.008 that of Earth, a surface gravity of 0.134 g, an escape velocity of 2.025 kilometers per second, 0.1810 that of Earth, and a surface pressure about one trillionth (0.000000000001) that of Earth.
4) Callisto, a moon of Jupiter, has a mass 0.018 that of Earth, a surface gravity of 0.126 g, an escape velocity of 2.440 kilometers per second, 0.2181 that of Earth, and an atmospheric pressure of about 0.00000075 Pascals.
5) Ganymede, a moon of Jupiter, has a mass 0.025 that of Earth, a surface gravity of 0.146 g, an escape velocity of 2.741 kilometers per second, 0.2450 that of Earth, and an atmospheric pressure of about 0.000001 Pascals.
6) Io, a moon of Jupiter, has a mass 0.015 that of Earth, a surface gravity of 0.183 g, an escape velocity of 2.588 kilometers per second, 0.2286 that of Earth, and a maximum atmospheric pressure of up to 0.0003 Pascals.
7) Pluto, the dwarf planet, has a mass 0.00218 that of Earth, a surface gravity of 0.063 g, an escape velocity of 1.212 kilometers per second, 0.1083 that of Earth, and a surface pressure of about 1 Pascal, about 1,000,000th to 100,00th that of Earth.
8) Triton, the moon of Neptune, has a mass 0.00359 that of Earth, a surface gravity of 0.0794 g, an escape velocity of 1.455 kilometers per second, 0.13007 that of Earth, and a surface pressure of about. 1.4 to 1.9 Pascals. The thin atmosphere of Triton is dense enough to have detectable winds.
9) The Planet Mars, has a mass 0.107 that of Earth, a surface gravity of 0.3794 g, an escape velocity of 5.027 kilometers per second, 0.4494 that of Earth, and an average surface pressure of about 610 Pascals.
10) The Planet Earth, has a mass 1.000 that of Earth, a surface gravity of 1 g, an escape velocity of 11.186 kilometers per second, 1.000 that of Earth, and a surface pressure of 101,325 Pascals.
11) Titan, a moon of Saturn, has a mass 0.0225 that of Earth, a surface gravity of 0.138 g, an escape velocity of 2.639 kilometers per second, 0.2359 that of Earth, and an atmospheric pressure of about 146,921 Pascals, about 1.45 times that of Earth. Hypothetical lifeforms that could survive on Titan should be able to fly much easier than on Earth.
12) The Planet Venus, has a mass 0.815 that of Earth, a surface gravity of 0.904 g, an escape velocity of 10.36 kilometers per second, 0.926 that of Earth, and a surface atmospheric pressure of about 9,300,000 Pascals, or about 92 times that of Earth. Hypothetical lifeforms able to survive on Venus should be able to fly much easier than on Earth.
At the present time it is rather uncertain why the atmosphere of Titan is about a hundred billion times more dense than the atmospheres of the similar moons Callisto and Ganymede.
The original question asked about flying in the atmosphere of a planet with three times the surface gravity of Earth. Unless the planet was a lot denser than any known planet, it would have to be much more massive than Earth to have a surface gravity of 3 g.
Planets more massive that Earth but less massive that gas giants or ice giants are called super-Earths. A number have been discovered in other stars systems, but usually little is known about them except for their masses and/or diameters.
In general, super-Earths are defined by their masses, and the term does not imply temperatures, compositions, orbital properties, habitability, or environments. While sources generally agree on an upper bound of 10 Earth masses14 (~69% of the mass of Uranus, which is the Solar System's giant planet with the least mass), the lower bound varies from 11 or 1.94 to 5,3 with various other definitions appearing in the popular media.57
https://en.wikipedia.org/wiki/Super-Earth6
As a general rule, if other factors are equal (and those other factors could be very unequal), a more massive Earth like planet would probably have more water. Thus it is possible that many super-Earths have more water and thus possibly higher percentages of their surfaces covered with oceans.
Even planets which are entirely covered with water might evolve flying life forms. The super-Earth sized exoplanets Kepler-62e and Kepler-62f in the habitable zone of Kepler-62 might be totally covered with oceans.
But such speculation is hard to resist. For example, Borucki raised the possibility that the newfound "super-Earths" — worlds just slightly bigger than our own planet — could host winged organisms, even if both planets are indeed water worlds.
"At least in our ocean, we have flying fish. They 'fly' to get away from predators," Borucki said.
"So we might find that they have evolved — birds — on this ocean planet," he added, referring to Kepler-62e.
Water worlds are unlikely to host technologically advanced civilizations like our own, Borucki and other researchers said, because any lifeforms that take root there would not have easy access to electricity or fire for metallurgy.
But if Kepler-62e or f has some dry land, Borucki said, the story could be different. The relatively high gravity of both exoplanets, however, might make the evolution of large bipedal organisms such as humans unlikely.
https://www.space.com/20728-new-alien-planets-oceans-life.html5
But:
An ocean world's habitation by Earth-like life is limited if the planet is completely covered by liquid water at the surface, even more restricted if a pressurized, solid ice layer is located between the global ocean and the lower rocky mantle.[49][50] Simulations of a hypothetical ocean world covered by 5 Earth oceans' worth of water indicate the water would not contain enough phosphorus and other nutrients for Earth like oxygen-producing ocean organisms such as plankton to evolve. On Earth, phosphorus is washed into the oceans by rainwater hitting rocks on exposed land so the mechanism would not work on an ocean world. Simulations of ocean planets with 50 Earth oceans' worth of water indicate the pressure on the sea floor would be so immense that the planet's interior would not sustain plate tectonics to cause volcanism to provide the right chemical environment for terrestrial life.[51]
https://en.wikipedia.org/wiki/Ocean_planet#Astrobiology7
So there are many factors to consider while designing your fictional planet.
