The story was circulating in early 2000s, so presumably it happened in 1990s. Kontsevich, it goes, opened a lecture course on mirror symmetry with:"This course is about two categories. One I will not define, the other nobody knows how to define, and we will be proving that they are equivalent".
I like the story because it highlights the informal side of mathematics, and counters the impression that what mathematicians do is logically deduce theorems from ready made definitions and axioms. But did it actually happen? There is no doubt that Kontsevich was giving lectures on mirror symmetry with categories in them, categorical mirror symmetry is his field after all. But did he say anything resembling the quote, or was it embellished like Eric Temple Bell embellished Cole's "talk without saying a word" about Mersenne numbers.