I think these kinds of constants are very normal (and very cool) in these kinds of problems. "e" shows up a lot. For example, the probability that a given bin is empty is the probability that every ball misses it, or (1 - 1/n)^n --> 1/e.
Another interesting way to formulate it is as a differential equation in the number of empty buckets as a function of time. Given k empty buckets, the probability of picking an empty bucket is k/n. For large values of n and k, we can pretend it's a continuous and not-particularly-stochastic differential equation dk/dt = -k/n, with k(0) = n.
That's classic exponential decay, the solution being k = ne^(-t/n). Thus k(n) = ne^(-1).
