What if there's a star inside a black hole?

There is a question that often comes up at every public talk I do, and I have never had a good answer to it—well, not an entirely confident one. What's inside a black hole? The textbook reply is a singularity. Cross the event horizon, and everything falls inward to a point of infinite density where our equations return nonsense. It isn't so much an answer as an admission that we've run out of physics.