TITLE: Expected maximum number of stones in a box
QUESTION [6 upvotes]: We have $N$ boxes and $N$ stones. We take a stone, put it into a randomly selected box, proceed to the next stone.
In the end all the stones are in the boxes. Some boxes may be empty, some may contain several stones.
Let $n$ be a maximum number of stones in one box in the end of this experiment.
What's the expected value of $n(N)$?
My intuition says that $n$ must grow with $N$, and computer simulations confirms this, but the growth is much slower than the intuition suggested.

REPLY [2 votes]: You're looking for the maximum load on putting $n$ balls into $n$ boxes.    The Wikipedia article (https://en.wikipedia.org/wiki/Balls_into_bins) gives that with high probability, the maximum load is ${\log n \over \log \log n}(1 + o(1))$, and  sources are cited there.