If a white dwarf compresses to the limit of electron degeneracy, and a neutron star compresses to the limit of neutron degeneracy, what does a black hole compress to the limit of?
In classical General Relativity, there is no limit to the compression in a black hole, hence you get a singularity. However, many astrophysicists feel that's unphysical, and that a theory which unites General Relativity and Quantum Mechanics will impose some kind of limit, perhaps something connected with the quantization of spacetime itself.
We don't have a working theory of quantum gravity, so at this stage we don't exactly know what happens in the core of a black hole. OTOH, we're fairly confident that the core has to be very small, since quantum gravity effects probably don't kick in until a scale much smaller than the size of an atom, and probably smaller than a proton, somewhere around the scale of the Planck length.
As far as current physics knows, nothing. This is the reason why it is commonly thought that a singularity exists in the middle of a black hole.
However, singularities are also thought to be non-physical, so there is most likely something else inside a black hole — we just don’t have the science to describe it just yet.