The answer to the question as stated (maximum of row elements) has been solved in Extreme statistics of complex random and quantum chaotic states, see also this MO posting:
$$\int dU \max_j |U_{1,j}|^2 =\frac{H_d}{d},$$
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).
The maximum of all matrix elements is more difficult and only large-$d$ asymptotics is known, see Maxima of entries of Haar distributed matrices (alternative link).
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.