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Titan has a thick atmosphere, as dense as Earth, and it not lose it.
What is the key? It is very cold and it has molecules too heavy to reach escape velocity at that temperature.

Could we have a moon at the Titan's atmosphere pressure but at Earth temperatures and breathable?

To the question of temperature, some methods could been proposed, like the active towers. The Earth's temperature is required in the layers near the ground, where they are living beings, while the loss of the atmosphere occurs in the outer layer, where the molecules are unlikely to collide with each other, and it could get up escape velocity and lost forever.

This issue can become so deep that it could lead to the development of a future branch of geoengineering. About how the heat is distributed by the layers of the atmosphere, and as radiation, convection and conduction between the gas creates the gradient.

Assume for a moment that this is not a problem. Ganymede dense atmosphere, for example, would isolate the heat generated by light from artificial fusion reactors. Escaping most as radiant heat, the last layer being the more "transparent" to these, so that the heat escape transparently maintaining very cold the upper.

The second is more complicated. Molecules? Heavy? A "unsolvable" problem occurs. Heavy molecules tend to be lower in the atmosphere, so that would probably unbearable for earthlings. And the necessary O2 and N2 would remain light, rising up and losing equally.

Here is an idea that has been haunting me. What if we make very large molecules ... heavy, but simultaneously form a very light gas?
Is that possible?

The first question you must understand is that the gas density depends on the space between molecules, not only of these. These molecules are separated from one another by various fundamental forces, creating a "rejection space". Is your weight, divided into the space, which will form the density. Another thing is that the molecules usually tend to create a similar space, since the effective space is much larger than the molecule, so they tend to create similar areas. So, normally, we could considerate that density depends only of molecular weight.

But this need not be so.

Imagine a sphere totally made of graphene. The area would be very small. I'm talking about hundreds of thousands of carbon atoms, not more. Inside, just empty space. Its effective space is huge, because they are areas "impenetrable". The number of carbon atoms grow with surface. The effective space of the "molecule" with the volume. With an area of sufficient size, density would be lower even than the less dense molecule ... hydrogen.
Yet this "molecules" would be very heavy! With lots of atoms.

So, these molecules tends to be at the top of atmosphere, but at the same time, it would be too heavy to reach scape speed. It could be a wonderful tool to make dense atmosphere relative stable in time. When a "molecule" of this kind breaks, it probably turns into a broken ball more dense and drops to the low atmosphere.

It may have defects. It is possible that the material proposed (graphene) did not have the right properties. For example, you may not be able to sustain its vacuum state for long because other molecules could pass through the graphene using tunnel effect. But it is possible that may exist other alternatives for the same purpose. For example, with larger areas, although some molecules enter the density molecules take longer to go down. The issue is not that they are always empty, but its state of lighter molecules that hydrogen could be sustained long enough. The location of a graphene layer could perhaps use a multilayer model with some technology that would allow these layers remain in a stable state.
The question would be to have these "balls" lighter than hydrogen stable long enough to need little repleshing.

I think that you are very optimist. Passive or "medium passive" methods like use special gasses on little quantities to generate a possitive feedback that could need centuries only to wait to get the correct temperature. But generete an little percent of gas ammount into an atmosphere, is huge for us. For example, change the CO2 in Earth Atmosphere to change the little quantity of CO2 that out atmosphere contains have required a century, massive process of burning carbons that is easy because generate energy, don't consume it and don't require advanced technology.

And it was burning is all our machines, the machines that work for a population of billions of humans.

Although CO2 have been generated as a waste it could show us, how difficult is any kind of planetary engineering that is some orders of magnitude bigger that change a little the composition of out atmosphere.

With current technology, it could require from thousands to hundred of thousands years to complete a terraforming. Perhaps Mars could be an exception and, if all is in the better case, it could be completed in some hundred of years. We need more investigation to check it.

But things like remove all carbon from venus atmospheres is a lot more complicated.

I think that future technologies could change that, but bring this technologies to reality need time. Only make fusion energy feasible requires now decades, and fusion will be necessary only to make a serius colonization in centuries instead millennia. So terraforming is far, far away.
Perhaps, future technologies like selfreplication machines could change some orders of magnitude to make terraforming feasible. I hope it. But the numbers are really big in any case.

The total automatization of tunnel construction in Earth is now under investigation to make superstructures like build larger and deepers tunnels where is difficult to grant the air at construction time.

Moon is now without atmosphere. I'm sure that a structure could be constructed rotating over other pushed over a superconductor (to avoid friction) and using electromagnets to gain the speed loose.

The structure would be like a curved cone to make the floor always perpendicular to the sum of real gravity and centrifugal force.

This structure could be constructed in a subterranean cave to avoid dust and have greater protection.

Probably too little, but Argon is far more abundant that Xenon or Krypton (Argon could be generated by fusion while heavier elements that Iron are not).
I suppose that with argon, if is enough heavy for moon, we could recolect it in different places of the Solar System.

With viable and economic transmutation, argon abundance would be guarranteed because chlorine is abundant.

Perhaps the channel of comunication is not appropriate.

Some people believe that electromagnetic radiation interfere too much with living beings so, if a new faster method (faster than light particules for example) is developed and it could be easily adapted to small devices, electromagnetic comunications could be deprecated by advanced civilizations.

Yeah... You are right.
It's too much Iodine. I think that the fusion was not the problem in the future with a little more advancement using other fuels that generates little energy per neutron produced.

But iodine is the big problem. Altought it is more abundant that xenon, create this atmosphere requires a large amount of mass.

Ok... which alternatives we have?

Requisites:
- Heavy molecule, enough to be retained on Moon and reach 1 atmosphere of pressure
- Very stable. It must tolerate great temperature without break (at high altitude, the temperature could change a lot)
- Not a powerful greenhouse gas. We don't want another Venus
- Chemically neutral (we don't want a corrosive atmosphere)
- Enough abundance in Solar System
- As little toxic as possible

Heavy noble gases are ideal except in the abundance.

Another alternatives?

Thanks.

I know but this is the great part of this.

Xenon could be generated as a byproduct of neutronic fusion like deuterium-deuterium fusion.
Neutronic fusion require some materials shielding the reactor. If we used compounds that contains iodine, it will be converted into xenon.
Creating xenon would not consume energy. Xenon would became a free gift from using neutronic fusion.

I think that Xenon atmosphere is a great idea.

Not in a terraforming sense. Moon is too small to retain a breathable atmosphere. But a dense atmosphere would make Moon easier to colonize.

The atmosphere is a good shield for small meteorites. Allow to transfer heat from the cool side to the hot side of Moon. Machines work better with air because heat could be dissipated throught convection or conduction plus always existent radiation.
Domes will be a lot more secure if the pressure is near or the same inside and outside it. Leaks of air will be reduced too (even more if outside is at more pressure that inside, because the air push into). Because xenon is not toxic, possible leaks are not critical if enough oxigen is retained inside the dome.

Without atmosphere, transparent and slight domes are too much difficult and we will use underground colonies the first times. But this colonies are closed boring places, and far less attractive to live in.

Where xenon could be taken? Well, make an atmosphere, like terraforming, is always a long time task.
Xenon is probably almost always in form of gas, so probably we could not found it in enough amount outside planets. So, or we collected from gigant  planets (very deep in atmosphere, so really hard to recover it) or we found it on unpredictable places, or it will not be enough xenon to make this artificial atmosphere.

Another possibility could be make xenon artificially through transmutation of iodine with neutron radiation that I hope could be easily created with fusion reactors were became available.

Although iodine is a rare element too, because is found in compounds, exists a greater probability to found enough large deposits in some place of the solar system in solid forms, perhaps on the meteorites of the asteriod belt, that is very much easy to catch that extract from a gigant planet.