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 $X_i=phicirc f^{i-1}$ 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 $mu(M_nleq u_n,textrm{i.o.})$ 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.
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