We can imagine n uniform random variables as an n-dimensional hypercube.
There is a simplex with an "orthogonal corner" of all zeros within this
hypercube which separates the region where the n variables sum to less than 1
from the region where they are greater than 1. The volume enclosed by this
simplex is 1/n!.
With a little bit of algebraic manipulation, we then see that the expected
number of draws required is
1/1! + 1/2! + 1/3! + ... + 1/n! + ... = e