A singularity may be indefinitely dense, meaning all its mass is concentrated into a single point in space, with no surface or volume, or something completely different.
It's hard to imagine everything in the whole cosmos — every star, nebula, galaxy; every atom, electron, and proton — all squeezed together into one infinitely dense blob.
It doesn't seem to make any sense that you can have a defined space with a finite area, like Great Britain be surrounded by an infinitely long perimeter.
After arriving in England, when Chandrashekhar showed his results to Arthur Eddington, the astronomer couldn't believe that a star could become infinitely dense.
In the 60s, Stephen Hawking and Roger Penrose predicted that an even denser star would contract into a point of infinite density and hence spacetime curvature, a point where all known laws of physics break down.