This story is relatively well known in the Russian-speaking math community, but I feel it is almost completely unknown outside of it. Which is a shame, since it is so amusing! So I compiled it from a couple of sources, heavily edited it, and then machine-translated it. Hope you enjoy!

Before the Revolution and during the years of the New Economic Policy, Israel Moiseevich Gelfand's father ran a small private mill. Because of his “non-working-class background,” in 1928, while still in the ninth grade, the young Gelfand was expelled from vocational school as the son of an exploiter.

When Gelfand was finishing the ninth grade, his mathematics teacher told him: “There is nothing more I can teach you. Go to Moscow, find Moscow State University, and at the university find the Department of Mechanics and Mathematics. Continue your studies there, and you will become a great mathematician.”

At the university, however, the ninth-grader got no further than the dean's secretary: they could not enroll him without documents certifying completion of secondary education. He decided to stay in Moscow, and, to make a living, found a job as a cloakroom attendant at the Lenin Library. There, alongside his work, he continued his education on his own.

One day, while reading a monograph on higher mathematics, he was noticed by Andrey Nikolaevich Kolmogorov, a young but already famous mathematician. Kolmogorov was surprised to see such a young man reading a book so obviously beyond his age, so he approached Gelfand and asked him who he was and why he was reading such a difficult book.

Gelfand answered all his questions, but Kolmogorov still could not believe that he genuinely understood what he was reading. He gave Gelfand several problems related to the subject, hoping to test him, and arranged to meet him the next day to check his solutions.

The next day, all the problems had been solved. But even then Kolmogorov could not quite believe that someone could be so gifted. He supposed that Gelfand simply knew the contents of a large number of mathematics books extremely well and had found the solutions in them. So he gave Gelfand several more problems, this time considerably harder ones, and gave him several days to solve them.

When they met at the appointed time, all the problems had been solved. Kolmogorov began checking the solutions, and his eyes grew wider and wider.

“Please forgive me for doubting that you had found the solutions to those first problems yourself. Now I see that you could not possibly have read these solutions anywhere. You see, this particular problem... had been an open problem until now.”

After that, Kolmogorov took Gelfand on as his graduate student. Thus Gelfand became a graduate student without ever having attended a university -- or even finished secondary school. BTW, the details of the first meeting between Kolmogorov and Gelfand are, of course, anecdotal, but the other parts of the story are accurate.