What are the Conflicting Predictions of General Relativity & Quantum Mechanics?

I see a lot of questions in various sites about why the 2 theories are or aren't incompatible, I'm satisfied as to why that's the case.

However it has been mentioned that both theories make predictions about phenomena that contradict or are incompatible, and I've been unable to find any examples.

What are the conflicting/contrary/incompatible predictions made by General Relativity versus Quantum Mechanics? Or are the claims false?

Answer

As Mitchell Porter said, we need more than just the absence of self-evident contradictions. We need a theory that encompasses both quantum mechanics and general relativity. The most straightforward, naive union of these frameworks produces a theory that is "nonrenormalizable" – predicts all quantities to be equal to a finite number plus infinity (many types of infinities arise).

String theory is the only known reconciliation of general relativity and quantum mechanics and chances are high that this status won't ever change.

There's of course no contradiction between "appropriately reduced in reach" general relativity and "appropriately tamed" quantum mechanics – after all, to a certain extent, both of these frameworks have been established so there has to exist a more accurate theory that agrees with all the established insights.

However, there are contradictions between quantum mechanics (which seems perfectly exact and valid) and classical general relativity believed literally. For example, classical general relativity paints a black hole as a perfectly determined, unique state of the spacetime. It carries no entropy because it has "no hair" according to classical general relativity. According to quantum mechanics, this can't be the case. A black hole has to carry and does carry a huge entropy – in fact, a greater entropy than any other localized object of the same mass – which is needed for the second law of thermodynamics to hold (entropy has to increase in time) and which is needed to "preserve the information" about the initial state, something that is required by "unitarity" in quantum mechanics. So some "tunneling" of the information from the interior has to be possible.