We consider the number NθA(θ) of eigenvalues eiθj of a random unitary matrix, drawn from CUEβ(N), in the interval θj∈[θA,θ]. The deviations from its mean, NθA(θ)-E[NθA(θ)], form a random process as function of θ. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher-Hartwig conjecture supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any β>0. It exhibits combined features of standard counting statistics of fermions (free for β=2 and with Sutherland-type interaction for β≠2) in an interval and extremal statistics of the fractional Brownian motion with Hurst index H=0. The β=2 results are expected to apply to the statistics of zeroes of the Riemann Zeta function.