The answer to the question as stated (maximum of row elements) has been solved in <A HREF="https://arxiv.org/abs/0708.0176">Extreme statistics of complex random and quantum chaotic states</A>, see also this <A HREF="https://mathoverflow.net/a/298776/11260">MO posting</A>: 

$$\int dU \max_j |U_{1,j}|^2 =\frac{H_d}{d},$$

with $H_d=\sum_{j=1}^d 1/j$ the harmonic number. For large $d$ this tends to $(1/d)\log d$. The complete probability distribution of the row-maximum is known (Gumbel distribution).

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The maximum of _all_ matrix elements is more difficult and only large-$d$ asymptotics is known, see <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 $W_d^2\rightarrow (2/d)\log d$ in probability for $d\rightarrow\infty$, so twice as large as for the row-maximum.