  The probability distribution of the maximum $M_t$ of a single resetting
Brownian motion (RBM) of duration $t$ and resetting rate $r$, properly centred
and scaled, is known to converge to the standard Gumbel distribution of the
classical extreme value theory. This Gumbel law describes the typical
fluctuations of $M_t$ around its average $\sim \ln (r t)$ for large $t$ on a
scale of $O(1)$. Here we compute the large-deviation tails of this distribution
when $M_t = O(t)$ and show that the large-deviation function has a singularity
where the second derivative is discontinuous, signalling a dynamical phase
transition. Then we consider a collection of independent RBMs with initial (and
resetting) positions uniformly distributed with a density $\rho$ over the
negative half-line. We show that the fluctuations in the initial positions of
the particles modify the distribution of $M_t$. The average over the initial
conditions can be performed in two different ways, in analogy with disordered
systems: (i) the annealed case where one averages over all possible initial
conditions and (ii) the quenched case where one considers only the
contributions coming from typical initial configurations. We show that in the
annealed case, the limiting distribution of the maximum is characterized by a
new scaling function, different from the Gumbel law but the large-deviation
function remains the same as in the single particle case. In contrast, for the
quenched case, the limiting (typical) distribution remains Gumbel but the
large-deviation behaviors are new and nontrivial. Our analytical results, both
for the typical as well as for the large-deviation regime of $M_t$, are
verified numerically with extremely high precision, down to $10^{-250}$ for the
probability density of $M_t$.
