most moons

Pretty much every moon but Titan. Titan, however, would be excellent for heat dissipation. Long before generative AI was even a thing, scientists have speculated that Titan would be the perfect place for datacenters because low-temperature computation is so much more efficient.

Of course, building a datacenter on Titan would be a several-hundred-trillion dollar endeavor, so... good luck bootstrapping your way into that industry.

None of the moons in our solar system have atmospheres. Earths moon is too small to hold on to an atmosphere, and the Galilean moons of Jupiter are too cold for an atmosphere, the gases just freeze.

The best place would be either a space station in low earth orbit or of the L4 or L5 point. The data issue would be the problem though I suppose you could just use the data centres for training but not for active processing but then you would need to build data centres on earth for that.

Given that you're going to build the earth data centres anyway you might as well do all of the processing on earth at the same time.