Some assumptions and thoughts
Air mix is same as Earth for oxygen and nitrogen. It is the partial pressures of these gases causing toxic reactions which would be fatal, not the physical force exerted by the pressure. This is provided the faller does not have any unreachable air cavities inside their body which would cause physical harm when slowly squashed to a smaller size. Most of the human body is relatively incompressible, being made of water, and the purely physical resistance to high pressures of the human body goes well beyond where a normal air mix would be highly toxic.
Gravity is somehow around 1g, and does not change significantly over the distance of a fall (as you will see below in general the falling distance is small compared to planet radius, so it's not too bad an assumption). Some gas and ice giants do appear to have this gravity in their outer layers, but Jupiter has for example 2.4g, where the numbers would be quite different (and make it quite hard to achieve anything)
We start the fall from a typical sky-diving height, and then add on smaller "layers" that are roughly equivalent to one earth atmosphere. Each layer is thinner than the last, because it is denser. The extra density also reduces the terminal velocity.
I quickly researched the pressure at which normal air becomes toxic to humans, and found recommended safety limits for diving with normal air at 66 metres of water, which is 7 bars (1 bar == 1 normal atmosphere). This is considered safe for long-term operating. For the duration of a temporary drop we can probably double that and still have plenty of room before short-term toxic effects become a problem.
I am ignoring possible effects of temperature changes with depth. I have absolutely no idea how significant they would be, or even sure in what direction. My gut feel is that temperature would increase with pressure, and could become a serious threat in addition to the toxicity. A higher temperature would also reduce the density required for a specific pressure, making each layer thicker, increasing time possible for the drop, provided it didn't get too hot. For comparison, on Earth, a height difference of 1000 metres in the densest part of the atmosphere corresponds to a temperature difference between 5 and 10 degrees Celsius. At that rate it could get seriously hot on a relatively shallow drop.
The density of air is proportional to the pressure. At sea level, 1 millibar of pressure change is roughly equivalent to 9 metres distance. But you do need to now include the fact that the pressure is higher, so the next one millibar going down might be in 8.99 metres. When you ignore temperature, then the relationship between pressure and volume is covered by Boyle's Law. - which means the same weight of gas (that applies yet more pressure further down) is in a smaller volume. However, to calculate pressure you effectively take a fixed-unit width "column" of air to calculate the weight. Which means air at twice the pressure is twice a dense and creates the same amount of pressure in half the height.
Rough guesstimate
I cannot be bothered with the maths for a smooth curve*, seeing as we're also ignoring some other physics. So we can approximate the fall as follows:
Starting at 1 atmosphere, the unlucky person falls 9000 metres with a terminal velocity of $60 ms^{-1}$, taking 150 seconds.
The next section is at 2 atmosphere. It is only 4500 metres deep, but terminal velocity $v$ is proportional to density $\rho$ as $v \propto \frac{1}{\sqrt \rho}$, so speed is $42 ms^{-1}$, taking 106 seconds.
Next section is 3 atmospheres. 3000 metres deep, velocity $35 ms^{-1}$, taking 87 seconds.
We can see a pattern here, each deeper part of the atmosphere takes a little less time to cross. It will take quite a while to reach what is considered safe diving distance pressure in air at 1g. But once at that depth, problems with toxicity will start happening faster and faster.
Doing the sums up to completing layer 7, I get roughly 600 seconds, or 10 minutes of sky-diving, over a distance of about 23 kilometres. This is not much distance at all, compared to the depth of our "full" atmosphere. That's because there are very large distances spanned by ever more tenuous gases. Once you get into breathable sections of air, it is not very deep at all.
Interestingly the character would be travelling at 80 km/h (or 50 mph), which is below the top speed of an experienced cliff diver, and roughly equal to speed reached from a fall from 30 metres in our normal atmosphere. Our faller could survive impact with a little luck or skill, and should be able to cope long term with the need to breathe at least.
Going as far as the "risky" layer 15, I get 960 seconds, or 16 minutes, and a distance of almost 30 km. At this pressure, the character will have been breathing in increasingly dangerous partial pressures of nitrogen and oxygen for the last 6 minutes. I think nitrogen narcosis would strike earlier out of the two, but I have not been able to find accurate timescales on which it would occur.
Our intrepid falling character could go further perhaps. But every minute they would be breathing a highly toxic mix of gases. They would likely already be suffering ill effects, confusion, hallucinations. They would quickly black out and die. I cannot give you a time for those effects to occur, or a point of no return.
Including temperature effects
Looking at a fact sheet for Jupiter there is a temperature profile of that planet which shows a 2 degrees Celsius warming per kilometre in the gas giant's atmosphere, at the cloud-tops level and below. So assuming you start close to zero Celsius at the top, a person could drop maybe 20km before heat stress started being a bigger problem than poisoning. But you could also maybe handwave in a gentler gradient to allow a longer drop. You have the choice of two nasty ways to threaten your character.
* You can get better accuracy on this estimate, and still avoid getting into calculus, by making each layer smaller. I have done so, using several thousand layers with much smaller changes between them. I have also checked the calculations against an online pressure calculation tool and got agreement within a few tens of metres for depths at pressures of 2000, 3000 and 4000 bars. The difference is on the order of 10% shorter times and distances, probably less than the effect of ignoring temperature, and within range of changing other variables such as composition of the atmosphere, strength of gravity etc. So I have left the original numbers in from the crude estimate, because it is easier to explain.
