Expected maximum bin load, for balls in bins with equal number of balls and bins
The correct approximation here is Poisson. The occupancy of each bin is distributed roughly $P(1)$ (Poisson with expectation $1$), and tail bounds for Poisson random variables show that with probability $1-1/n^2$ (say), the variable is at most $O(\log n/\log\log n)$. A union bound shows that with high probability, all bins contain at most $O(\log n/\log\log n)$ balls.

A good resource on balls and bins is "Balls into Bins" – A Simple and Tight
Analysis by Raab and Steger. They use elementary methods to a range of $m$ (balls) and $n$ (bins) that covers the classical case $m = n$.