In General Relativity, the way to get gravity so strong that not even light can escape is with a singularity: a point of infinite density. So, either this infinity physically exists, and maybe we'll understand how better, or General Relativity may be incomplete: a model that works well most of the time, but doesn't represent reality correctly at the extremes of heavy mass and small space.

Or at least that's how I understand it. This has more info: https://physics.stackexchange.com/questions/18981/why-singularity-in-a-black-hole-and-not-just-very-dense#18987

This is similar to the ultraviolet catastrophe. Physicists predicted black body radiation using their current physical models, with high accuracy at low wavelengths of light, but at high wavelengths, the predictions diverged towards infinity, which disagreed with measurements.

(Source: Wikipedia)

Breakthroughs in quantum physics later reconciled theory with measurements.

One big difference with black holes is we cannot yet measure the actual density in the interior of the black hole. We just have the prediction that there is a point of infinite density.

Any physicists around here may have a better understanding than me.