@CortAmmon This is a decent popularization, but I think it's doing a common unsatisfying thing with quantum gravity. The numbers "balloon and explode off towards infinity" in just about all quantum field theories; gravity is not special in this sense at all. The problems with quantum gravity are more subtle, and are covered elsewhere on this site.

@knzhou My understanding was that the explosions off to infinity could be dealt with via renormalization, but the curvature of space from gravity distorted things such that the math of renormalization no longer worked.

@CortAmmon You can still absorb all of the infinities by renormalization for quantum gravity. The problem is that there is an infinite number of infinities, so your theory ends up having to be described by infinitely many parameters. This is still okay because all but a few of these parameters don't matter in the low-energy limit, where only one (the Einstein-Hilbert term) matters. But as you go to higher energies they all start to matter, so you can't predict anything.
In fact this situation comes up pretty often in physics, and it's been resolved for subparts of the Standard Model several times in the past. You just find a larger theory to embed the original theory in, which actually is predictive. That's the modern perspective of effective field theory. It is true that there are older sources that speak only in terms of divergences, but these were from an era where renormalization wasn't as well understood.