This problem has been considered in the large-$d$ limit by <A HREF="https://link.springer.com/article/10.1007/s00440-004-0376-5">Maxima of entries of Haar distributed matrices</A> (alternative
<A HREF="http://users.stat.umn.edu/~jiang040/papers/haar1.pdf">link</A>).

If $W_d$ is the maximum matrix element in absolute value of a Haar-distributed $d\times d$ unitary matrix, then $W_d^2\rightarrow (2/d)\log d$ for $d\rightarrow\infty$.