Suppose I want to have a moon with more gravity than ours how large of a moon can I have and what effects would there be.
To be a body orbiting another (and not otherwise), your moon has to be at most the same mass of our planet. Since it is the mass that matters, and not the size, you can make your moon as large as you want, taking in account that the moon.mass/(moon.size/2) must be higher than the planet.mass/(planet.size/2) -- so bodies would be heavier at surface. As you ask for the optimum size, let us take example scenarios:
A) The two cosmological bodies have the same mass (there is, not a 'moon' system properly, but a reciprocal orbit); then gravity would act strongly on the surface of the smaller body.
B) The moon is half the mass of the planet; then you would weight the same on surface if the moon size is half the planet size; anything smaller that that would have stronger gravitational pull.
C) The moon weights a fraction the mass of the planet. Say 1/87 (that is approx. our real scenario): you would weight the same on surface if the moon size was 1/87 the size of Earth, (that would be approx. 146 km diameter, but the moon actually has a 3474,2 km diameter). Smaller than 1/87 the planet diameter in size, bodies receive stronger pull.
So it all depends on how heavier to you want your moon to feel.
You want the moon to feel twice as heavy than the planet; the mass/size proportion of the moon must be at least 4 times the mass/size proportion of the planet.
If you want it three times heavier, the mass/size of the moon must be 6 times more. 10 times heavier: 20 times the planet's mass/size, you get the point.
Moon feels twice heavier than the planet: moon size is maximum = (moon.mass * planet.size) / (2 * planet.mass).
Moon feels three times heavier: moon size maximum = (moon.mass * planet.size) / (3 * planet.mass).
And so forth.
The OP needs to specify a number of details about what he wants. Do they want a story in which the "planet", or the "moon", or both, has life, possibly including intelligent life, or can both worlds be lifeless? How much more "gravity" does the OP want the moon to have compared to the planet?
Once the OP clarifies what is desired, someone might be able to calculate what is necessary for that to happen.
Part One of Two. A discussion of the requirements.
The OP failed to specify what they meant by the "moon" having more "gravity" than the "planet".
The surface gravity of an astronomical body determines how heavy people would feel on its surface and how fast and hard a falling object or person there would fall.
The escape velocity of an astronomical body, combined with the average mass and average temperature of gas molecules in the escape layer of its atmosphere, determines how long the astronomical body will be able to retain an atmosphere.
Since a number of science fiction stories - especially earlier ones - set on Earth's Moon have featured the discovery of living or extinct lifeforms, which requires liquid water and an atmosphere, the OP question might be based on a story idea for explorers from the fictional planet to find interesting things on the the fictional moon, interesting things which might require a past or even present atmosphere, and thus require that the fictional moon probably has a much greater mass and escape velocity than Earth's Moon does.
Or the OP question could be based on a story idea for explorers from a fictional planet to land on its fictional moon and find out that the fictional moon's surface gravity is much higher than that of the fictional planet, although their scientists should have already calculated the moon's surface gravity.
There are different formulas for calculating the surface gravity and the escape velocity of an astronomical body. And it seems that the surface gravity and the escape velocity of an astronomical body do not change in unison, so that it would be theoretically possible for an astronomical body to have, for example, an escape velocity slightly higher than Earth's and a surface gravity slightly lower than Earth's.
What is the dividing line between a large moon of a planet and a companion planet of that planet?
Some people have called the Earth and the Moon a double planet, even though Earth has about 4 times the diameter, 64 times the volume, and 81 times the mass of the Moon.
And some people have called Pluto and Charon a double planet, or since 2006 a double dwarf planet, even though Pluto has about 2 times the diameter, 8 times the volume, and 8.5 times the mass of Charon.
In my humble opinion, the fictional "moon" would have have to be a lot closer to the mass of the fictional planet than Charon is to Pluto in order to have either a surface gravity or an escape velocity that would be higher than that of the planet. If the difference in surface gravity or in escape velocity has to be really high in the story the fictional "moon" might even have to be much more massive than the fictional "planet".
So either the story will have to use a definition of the border between a planet and moon system and a double planet system which requires the two bodies to be very similar in diameter, volume, and mass to be considered a double planet, or else the story will have to be set in a double planet system and the characters will consider it a double planet system.
The way to make an astronomical body which is smaller in diameter and Volume, and/or in mass, have a higher surface gravity and/or escape velocity than another astronomical body with a larger diameter and volume and/or mass, is for it to have a much greater average density.
But there is a problem. The more massive an astronomical body is, the more its gravity and the pressure of the higher layers will compress the lower layers of the astronomical body. If the fictional planet is a life bearing planet that resembles Earth, it will be large enough that its gravity will have compressed its materials to an average density which is much greater than that material would have floating in tiny meteoroids in space.
If the fictional "moon" has less mass than the fictional "planet", its lesser gravity should should have compressed its material to a lesser average density than the material of the fictional "planet".
Unless the two astronomical bodies were made of different materials which have different average densities when not compressed, which can be compressed to different degrees. Thus a smaller or less massive body could be made of much denser materials.
If both the fictional "planet" and the fictional "moon" are small rocky bodies like the inner planets in our solar system, they would have average densities similar to those of the four inner planets and the moons and asteroids in the inner solar system.
Earth has a density of 5.514 grams per cubic centimeter, Venus 5.243, Mars 3.93, Mercury 5.427, the Moon 3.3464, Ceres 2.17, Pallas about 3.0, and Vesta 3.46.
If the planet in the story is a habitable planet with native intelligent life forms who develop civilization and and send explorers to their "moon", it should be massive like Earth and have a high average density like Earth. Thus it is easy to imagine that the density of that planet might be about 5.0 to 6.0.
And it is possible that a habitable Earth like planet could have a density less than 5.0. Maybe as low as 4.9, 4.8, 4.7, and so on. Maybe an Earth like planet with advanced life could have an average density as low as about 4.0. Maybe an Earth like planet with advanced life could have an average density as low as about 3.0.
And maybe a smaller astronomical body could have a much higher density than some Earth like planets. Maybe a density of 6.0, or 7.0, or 8.0.
The most common dense material in our solar system is probably iron. Other elements are denser but much rarer, or else more common but less dense. All the inner planets of the solar system have iron nickel cores.
Iron meteorites are meteorites that consist overwhelmingly of an iron–nickel alloy known as meteoric iron that usually consists of two mineral phases: kamacite and taenite. Iron meteorites originate from cores of planetesimals.2
Kemacite has a density of 7.9 grams per cubic centimeter.
Taenite has a density of about 8 grams per cubic centimeter.
So the iron nickel alloys in the metallic core of a planet will have a density of about 8 grams per cubic centimeter even before gravity compresses them.
So I can imagine a situation where a warm Jupiter or a warm Neptune, or a really giant super Earth orbited in the habitable zone of a star early in the formation of that system. And then another planet collided with it and both planets were shattered.
But the iron nickel core of the warm Jupiter planet, remained mostly intact, and stayed larger than the iron nickel core of Earth. Most of the planetary material was lost into space, but enough remained in orbit around the center of gravity of the planet to form a ring with several times the mass of he remaining iron nickel core. And that ring gradually condensed to form another astronomical body, one which had more mass than the iron nickel core, and became the new primary in a new double planet or planet & moon system. But the larger and more massive world was made of much less dense material and so had much less density than the remaining iron nickel core, which thus could have had a higher surface gravity and/or escape velocity.
or possibly there was a double planet system orbiting in the habitable zone, and the larger planet was struck by a wandering planet in the early history of the solar system. the larger planet may have lost all of its mass except for the iron nickel core, and most of that mass might have landed on the smaller planet and enlarged it, thus making what was once the smaller planet the larger planet, and what was once the more massive planet the less massive planet, or "moon".
Part Two of two: A really wild idea.
Involves small asteroids made of white dwarf star material, which is super dense. By piling on normal matter from normal asteroids on top of the white dwarf material, the surface gravity of the super dense asteroids can be reduced to the desired amount.
Thus an astronomical body could have very small diameter and volume but have such a high density that it could have an arbitrarily high surface gravity or escape velocity.
Unfortunately, if the "degenerate" matter from a white dwarf star is removed from the mass and gravity of the white dwarf star it will expand until it has the density of normal matter. So a tiny world made of white dwarf star material need to have the mass of a white dwarf star in order to say compressed and avoid decompressing with explosive force.
And the same goes for the even denser matter in a neutron star. If it is, for example, teleported away from the neutron star and its gravity, it will expand with explosive force to occupy a much larger volume until it has a normal density. So a writer can't use a small chunk of neutron star material to make their "moon" have a much higher gravity than normal, not unless they ignore science in their story.
Matter is even more condensed and dense within a black hole. In fact, theoretically all the matter within a black hole should be within an infinitely small and infinitely dense singularity at the center of the black hole. Since the center of the black hole and the singularity would be surrounded by the event horizon there would be no way to tell if that was correct.
Theoretically low mass black holes could exist, with masses much smaller than that of stars, planets, and moons.
So theoretically such a low mass black hole could exist at the center of a moon. And if the amount of matter on top of the event horizon of the black hole was large enough, the surface gravity and/or the escape velocity of the moon could be as much as is desired for a story.
Unfortunately, there two properties of such theoretical mini black holes that affect a story.
Mini black holes would emit Hawking radiation, and the smaller the mass of the black hole, the more radiation it would emit and the faster the black hole would lose mass. And the faster the black hole loses mass, the sooner it loses the last remaining bit of its mass in a big explosion.
So the mini black hole would have to be massive enough to not lose mass too fast and last for the duration of the situation in the story, which might mean lasting for billions of years.
Of course the mini black hole would also have to be massive enough for its mass at the center of the moon to raise the surface gravity and/or escape velocity of the moon to the levels desired in the story, and so it might have to be several times as massive as the ordinary matter surrounding it on the moon.
The other property of theoretical mini black holes would be that their gravity would attract particles and objects around them, which would tend to fall into the escape horizons of those black holes and add their mass to that of the black hole.
If the black hole is in the middle of an object made of ordinary matter, it will be where the density of matter is countless millions and billions of times greater than in interplanetary space and it will adsorb matter countless millions and billions of times faster.
So depending on how fast the black hole in the center of the astronomical body sucks up matter and how much matter the astronomical body has, the lifetime of the astronomical body could be seconds, minutes, hours, days, weeks, months, years, decades, centuries, millennia, etc., etc. before it is entirely sucked into the black hole.
In fact, it could take from between seconds and countless trillions of years for the black hole to totally adsorb the matter of the astronomical body.
The more massive the black hole, the stronger its gravitational pull on matter outside it will be, and the larger its event horizon will be, meaning that more matter will be able to fall into the event horizon within a unit of time. So as the black hole adsorbs more matter and grows more massive, it will adsorb matter from the astronomical body faster, which will enable it to adsorb matter faster, and so on and so on until eventually - after seconds to trillions of years - it will adsorb all the remaining matter at once.
So in order for the moon in the story to have much more gravity than it should, the black hole should:
1) Have a mass that is probably greater than that of the other matter in the moon.
2) have a mass large enough to not evaporate and explode before the period the story is set in.
3) A mass small enough to not adsorb all the other matter in the moon before the period the story is set in.
There should be a number of questions here about black holes in the centers of astronomical bodies.
LSerni's answer here: How can a small planetoid hold an atmosphere under artificial means?9
No doubt it would take a bit of calculation to find out if a black hole inside a moon could fit the story requirements and if so what the parameters would be.
In theory, the Earth could be a jovian moon, there is nothing physical forbidding that.
But size is only part of what makes a planet earth-like. You need:
1) Metals, lots of it. Due to rotation metals, being heavy, will be more present in the inner body then in the outer bodies. If a earth-sized moon formed around a jupiter-like planet it could be metal-starved and you need metals both for magnetic reasons and biological reasons. Maybe if the solar system in question is very metal-rich, the earth-sized moon formed very near the jupiter and was pushed to a farther orbit due to interactions with an even bigger moon that ended falling in the planet, you could have the metals you need. PS.: I have no idea what a iron-rich planetary cloud means to the central star. Iron is nuclear ash, it's fusion drains energy. There will be a lot of useless iron in the star, taking volume that should be filled with helium and hydrogen in the star's core. I have no idea what this iron poisoning will mean to the star.
2) Magnetic field. It will shield the world from the solar wind and keep it's atmosphere in place. That's why you need a lot of metals. Also, if you are orbiting a jupiter-like planet you have to deal with ionizing radiation in orbit and you need a magnetic field to avoid Chernobyl level radiation on surface.
3) Temperature. If it is too hot it won't be earth-like. And the problem is that your big moon will suffer a lot of tidal heating. Combined with a world with a lot of water that will give a very active tectonic plate system, which is a good thing. But combined with the solar heat and the greenhouse effect it could be too much heating. You the jupiter and it's earth-sized moon must be in a certain distance from the sun where the combination of volcanic heating won't overwhelm the life on the planet.
4) The problem of flood basalts. Flood basalt is the most dangerous cause of mass extinctions because they mess with solar input, damage ozone layer and fill the world with acid rains and too much nutrients in the sea. Your world, due to it's intense plate tectonics, will have a lot of problem with that. To give complex lifeforms a chance you will need alternate biological energy sources that don't rely on the sun and thrive in agressive volcanism. That means that the Black Smokers and the ecosystems around them will be as important as plants and plankton when it comes to feed the biosphere.