Did Gauss ever make a mistake?

You are seeing the down side of the "few but ripe" idea. He comes off badly in the episode with Bolyai. He did not publish his findings n a new, non-Euclidean geometry, over fears of shrill criticism from followers of Immanuel Kant. For the moment, I am going to assume that the word he actually used referred to Boethius; perhaps someone else knows. EDIT: no, it is an ancient insult, the citizens of Athens thought of their neighbors as dull and stupid.

The part I dislike is when Janos Bolyai's father, Farkas Bolyai, wrote to Gauss, asking his opinion of this new world. Gauss replied that he could not praise Janos, as he would then be praising himself, Gauss having done much the same thing decades earlier. I've always thought this a mixture of cowardice and lack of generosity. There really is a difference between doing a bunch of calculations and saying "If so and so happens, here are some rules that apply" compared with saying "Here is the actual thing."

Well, the full story is intricate, but Gauss's reply hit Janos very hard, and appears to have convinced him to drop mathematics; at least, he published no more.

Euler ranks higher than Gauss in my opinion. Having said that, I do not know if there is a recorded instance of Gauss making a mathematical error, but it's worth pointing out that the work of the 18th and early 19th century mathematicians was plagued by lack of rigor. Only later in the 18th century did the strides in analysis give mathematics a solid grounding.

how about errors of omission? Gauss was famous for not publishing monumental discoveries until after they were rediscovered.