  Many physical, chemical, and biological systems depend on the first passage
time (FPT) of a diffusive searcher to a target. Typically, this FPT is much
slower than the characteristic diffusion timescale. For example, this is the
case if the target is small (the narrow escape problem) or if the searcher must
escape a potential well. However, many systems depend on the first time a
searcher finds the target out of a large group of searchers, which is the
so-called extreme FPT. Since this extreme FPT vanishes in the limit of many
searchers, the prohibitively slow FPTs of diffusive search can be negated by
deploying enough searchers. However, the notion of "enough searchers" is poorly
understood. How can one determine if a system is in the slow regime (dominated
by small targets or a deep potential, for example) or the fast regime
(dominated by many searchers)? How can one estimate the extreme FPT in these
different regimes? In this paper, we answer these questions by deriving
conditions which ensure that a system is in either regime and finding
approximations of the full distribution and all the moments of the extreme FPT
in these regimes. Our analysis reveals the critical effect that initial
searcher distribution and target reactivity can have on extreme FPTs.
