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>>qqqqqu+(OP)
When two particles get closer, their mutual gravitational attraction increases. As the distance approaches zero, the force approaches infinity. In the limit of d -> 0, the energy released -> infinity. Obviously at some scale the notion of a point mass breaks down, but even quantum theory would be problematic if we think of a wave function as describing a probability distribution, wouldn't it? What's the "official" story on this?

>>bhk+An
What follows is very hand-wavy, and the renormalization sibling post may touch on it.

An answer is that the d->0 approaches infinity presumes a nice, continuous analytic function. If d->epsilon, you can't get to that singularity.

There was an equivalent problem in the E/M space with "The Ultraviolet Catastrophe" [1], which turned out to go away if you assumed quantization.

I'm not going to claim this is a perfect analog to the gravity problem, only that a lot of physics doesn't quite work right when you assume continuity. (The Dirac delta is a humorous exception that proves the rule here, in that doing the mathematically weird thing actually is closer to how physics works, and it required "distribution theory" as a discipline to prove it sound.)

[1] https://en.wikipedia.org/wiki/Ultraviolet_catastrophe

>>aberna+pp
This is definitely related, though not the whole story. Quantization did get rid of some infinities, but as GP kind of states, it also introduced others. My comment focuses on what we do for those.