I understand that the current estimate for the number of galaxies in the observable universe is about 100-200 billion. Is there anything in our understanding of physics and the evolution of the universe that would go against (or be inconsistent with) the idea of an infinite number of galaxies?
The observable universe has a finite volume, since it is of finite age and light can only travel so far in that time. As a result, it most certainly cannot harbor an infinite number of galaxies.
That said, there's no real reason the universe as a whole* can't be infinite. As far as our best measurements can tell, the patch we see is consistent with being "flat" - more like a flat sheet (if it were 2D) than the balloon it is often shown as. An infinite sheet, even with a finite age, very much can contain an infinite number of galaxies.
*One should be careful when talking about things outside the observable universe, but I'll do it anyway.