Let $B\_i(n,1/2)$ be independent identically distributed binomial random variables. I am interested in the asymptotic growth of the maximum of $n$ such random variables. In <https://math.stackexchange.com/questions/590880/bounds-for-the-maximum-of-binomial-random-variables> the asymptotics were shown to be bounded above by $\frac{n}{2}+ \sqrt{\frac{n}{2} \log\_e n}$ and numerically shown to be bounded below by $\frac{n}{2}+ \sqrt{\frac{n}{2.5} \log\_e n}$. 



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> Is it possible to derive the exact asymptotic behaviour for the
>  maximum of $n$ such independent and identical $B\_i(n,1/2)$ binomial random variables?
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> 


It seems (very) likely this is a solved problem but I haven't found where or by whom yet. I would be happy with a reference if that is available.



Asymptotics of the maximum of binomial random variables
What you need in order to compute the constant is to find the correct $x\_n$ so that
$$P(B\_1(n,1/2)>x\_n)\sim \frac{1}{n}$$
In this case it holds (by a precise asymptotics computation, based on normal approximation
and the fact that the variance is $n/4$)
that, with $\bar x\_n=x\_n/\sqrt{n}$,
$$P(B\_1(n,1/2)>x\_n)\sim \frac{C}{\bar x\_n} e^{-2 \bar x\_n^2}$$
So you should choose 
$$\bar x\_n=\sqrt{(\log n)/2}-\frac{\log\log n+C'}{8\sqrt{(\log n)/2}}.$$
With this choice, you will get that $P(\max\_{i=1}^n B\_i(n,1/2)<x\_n)\sim e^{-1}$.
This $\log\log n$ correction maybe explains the numerical simulations mentioned in the OP.
The fluctuations of the maximum are of order $1/\sqrt{\log n}$.


