Anthony, you wrote:
"Isn't there a "maximum" density for matter? I mean, stuff is composed of fundamental particles which supposedly have mass and take up space."
Hmm. But where does it say that fundamental particles take up space?? Electrons and quarks have no meaningful size.
Another poster said that you shouldn't take the infinite density seriously because GR and QM haven't been unified yet. Sure, that's a good point. But at what time after the Big Bang is that irrelevant? Ten times the Planck time? And so what density for matter would we have to deal with realistically. By the standard model of its origin, the universe was mighty dense a tiny fraction of a second after the Big Bang, and there's little reason to expect that "quantum gravity" will undo that.