For any positive integer $n$, let $[n] = \{1,\ldots,n\}$. Let $[n]^{[n]}$ denote the set of functions $f:[n]\to [n]$. For $f\in[n]^{[n]}$, we define the maximum accumulation $\text{macc}(f)$ by $$\text{macc}(f) = \max\{|f^{-1}(\{k\})| : k\in[n]\}.$$ (Note that $f\in[n]^{[n]}$ is bijective if and only if $\text{macc}(f) = 1$.)

Let $$E_n = \frac{1}{n^n}\sum_{f\in [n]^{[n]}} \text{macc}(f).$$ This is the expected value of the maximum accumulation when randomly given a member of $[n]^{[n]}$. I want to get a feeling of how fast $E_n$ grows as $n\to \infty$.

Questions. Do we have $\lim_{n\to\infty}\frac{E_n}{\log n} \in \;]0,\infty[$? Or is $\lim_{n\to\infty}\frac{E_n}{\sqrt{n}} \in \;]0,\infty[$? (If any statement is true, then the actual value of the limit would also be interesting to know.)