There was (to the best of our knowledge) a single big bang, since when energy is conserved. That gives us a total energy for the universe, which is large but finite, which puts an upper bound on the number of particles at any given time.

The big bang is just a theory that is obtained by extrapolating the current movement backwards. For all you know, the multiverse is infinite. And in fact so could the universe be -- the big bang could have happened much earlier and the universe could be much bigger. And when you say that there was only one big bang and there was a finite amount of energy, you are making too many asssumptions.