I am given $n$ objects and for $n$ times, I pick one of them with uniform probability and put it back after picking it.

For $k\in\{1,\ldots,n\}$ let $f_k$ denote the number of times that I have picked object number $k$. So we have $f_k\in \{0,\ldots,n\}$ for all $k$.

We consider $M:= \max\big\{f_k: k\in\{1,\ldots,n\}\big\}$, so we have $M\in\{1,\ldots,n\}$. We are interested in the expected value $E_n:= E(M)$.

Does $\lim_{n\to\infty} E_n$ exist? If yes, what is its value?