I'd throw the name Srinivasa Ramanujan into the genius hat.
Born in India and living in near complete poverty he taught
himself nearly a thousand years of western mathematics. That is to say, he didn't read a book and learn so much as derive most of what constituted advanced mathematics all by himself. He wrote a letter to Godfrey Hardy, a famous british mathematician, who arranged for a scholarship and had Srinivasa brought to England.
Once there he was able to toss out some really interesting (at least to a mathematician) work. The gaps in his education were unfortunately apparent since he didn't write his theorems in a way that most scientists found easily understandable or readable. Even so, some of his work is still poured over today. It's sad to think what he might have achieved if he had access to better resources as a child.
As to the OP and what others have written about prodigies and the like I'd venture to say this guy stands out given that what most people require a good deal of schooling for he had naturally wired into his brain.
Quote:
Ramanujan's knowledge of mathematics (most of which he had worked out for himself) was startling. Although almost completely ignorant of what had been developed, his mastery of continued fractions was unequaled by any living mathematician. He worked out the Riemann series, the elliptic integrals, hypergeometric series, the functional equations of the zeta function, and his own theory of divergent series. On the other hand, the gaps in his knowledge were equally startling. He knew nothing of doubly periodic functions, the classical theory of quadratic forms, or Cauchy's theorem, and had only the most nebulous idea of what constitutes a mathematical proof. Though brilliant, many of his theorems on the theory of prime numbers were completely wrong.
In England Ramanujan made further advances, especially in the partition of numbers. His papers were published in English and European journals, and in 1918 he became the first Indian to be elected to the Royal Society of London.
In 1917 Ramanujan contracted tuberculosis, but his condition improved sufficiently for him to return to India in 1919. He died the following year, generally unknown to the world at large but recognized by mathematicians as a phenomenal genius, without peer since Leonhard Euler (170783) and Karl Jacobi (180451).

Source: Encyclopaedia Britannica http://britannica.com/bcom/eb/article/1/0,5716,64161+1,00.html