Comment by coutud

Nope, there's an even simpler way.

Say f(x) is the expected value of the number of draws to get to value x. Then f(x) = 1 + integral_0^x f(x)

Hence f'(x) = f(x), so f(x) = c exp(x) The constant is of course 1, and f(1) = exp(1).