Homotopy type theory and Gödel's incompleteness theorems
HoTT "suffers" from Gödel incompleteness, of course, since it has a computably enumerable language and rules of inference, and we can formalize arithmetic in it. The authors of the HoTT book were perfectly aware of its incompletness. (In fact, this is quite obvious, especially when half of the authors are logicians of some sort).

But does incompleteness "impair" HoTT? No more than it does any other formal system, and I think the whole issue is a bit misguided. Let me try an analogy. Suppose you have a car which can't take you everywhere on the planet. For instance, it can't climb vertically up a wall. Is the car "impaired"? Of course, it can't get you to the top of the Empire State building. Is the car useless? Far from it, it can take you too many other interesting places. Not to mention that the Empire State building has elevators.