Carl Friedrich Gauss
Gauss was something of a child prodigy; the most commonly told story relates that when he was 10 his teacher, wanting a rest, told his class to add up all the numbers from 1 to 100. Gauss did it in seconds, having noticed that 1+...+100 = 100+...+1 = (101+...+101)/2.
Some idea of the range of his work can be obtained by noting the many mathematical terms with "Gauss" in their names. E.g. Gaussian elimination (linear algebra); Gaussian primes (number theory); Gaussian distribution (statistics); Gauss [unit] (electromagnetism); Gaussian curvature (differential geometry); Gaussian quadrature (numerical analysis); Gauss-Bonnet formula (differential geometry); Gauss's identity (hypergeometric functions); Gauss sums (number theory).
He nearly went into architecture rather than mathematics; what decided him on mathematics was his proof, at age 18, of the startling theorem that a regular N-sided polygon can be constructed with ruler and compasses if and only if N is a power of 2 times a product of distinct Fermat primes.