What Happens to Matter Inside a Black Hole's Infinite Density?

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If an object is infinitely dense, does this simply mean that there is no empty space within the object? I'm hung up on the fact that you can't possibly get denser than infinite density; what is stopping a black hole from getting even denser? What happens to atoms once they're under such intense gravitational pressure? !
 
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It is not believed that "infinite density" exists in the real world. It is a mathematical fiction that is the result of some model's calculations and is therefore taken as a sign that the model breaks down at the point where it gives infinity as a result.

For example, the most standard model of a black hole says there is a singularity at the center that has zero volume and all of the mass of the black hole and therefore infinite density, but what that really means is that we don't understand the black hole singularity. A successful theory of quantum gravity will likely give a different answer for what the singularity of a black hole is.
 

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