  Suppose $(f,\mathcal{X},\mu)$ is a measure preserving dynamical system and
$\phi \colon \mathcal{X} \to \mathbb{R}$ a measurable function. Consider the
maximum process $M_n:=\max\{X_1 \ldots,X_n\}$, where $X_i=\phi\circ f^{i-1}$ is
a time series of observations on the system. Suppose that $(u_n)$ is a
non-decreasing sequence of real numbers, such that $\mu(X_1>u_n)\to 0$. For
certain dynamical systems, we obtain a zero--one measure dichotomy for
$\mu(M_n\leq u_n\,\textrm{i.o.})$ depending on the sequence $u_n$. Specific
examples are piecewise expanding interval maps including the Gauss map. For the
broader class of non-uniformly hyperbolic dynamical systems, we make
significant improvements on existing literature for characterising the
sequences $u_n$. Our results on the permitted sequences $u_n$ are commensurate
with the optimal sequences (and series criteria) obtained by Klass (1985) for
i.i.d. processes. Moreover, we also develop new series criteria on the
permitted sequences in the case where the i.i.d. theory breaks down. Our
analysis has strong connections to specific problems in eventual always hitting
time statistics and extreme value theory.
