I remember that if there are $n$ boxes, and a ball is being thrown repeatedly into one of the boxes with uniform probability, then the expected number of throws before every box has a ball is approximately $nH_n$, which is approximately $n\log n$. The derivation goes as follows:
When we've filled $r<n$ boxes, the probability that we hit an unfilled box with each throw is $(n-r)/n$. The expected number of throws before we hit an unfilled box is $n/(n-r)$. By linearity of expectation, the expected number of throws before we fill all boxes is $\sum_{r=0}^{n-1}n/(n-r) = nH_n$.
What about the expected number of throws before every box has $k$ balls, where $k\geq 1$ is an integer? Is it going to be $kn\log n$.