  Triple stars and compact objects are ubiquitously observed in nature. Their
long-term evolution is complex; in particular, the von-Zeipel-Lidov-Kozai (ZLK)
mechanism can potentially lead to highly eccentric encounters of the inner
binary. Such encounters can lead to a plethora of interacting binary phenomena,
as well as stellar and compact-object mergers. Here we find explicit analytical
formulae for the maximal eccentricity, $e_{\rm max}$, of the inner binary
undergoing ZLK oscillations, where both the test particle limit (parametrised
by the inner-to-outer angular momentum ratio $\eta$) and the double-averaging
approximation (parametrised by the period ratio, $\epsilon_{\rm SA}$) are
relaxed, for circular outer orbits. We recover known results in both limiting
cases (either $\eta$ or $\epsilon_{\rm SA} \to 0$) and verify the validity of
our model using numerical simulations. We test our results with two accurate
numerical N-body codes, $\texttt{Rebound}$ for Newtonian dynamics and
$\texttt{Tsunami}$ for general-relativistic (GR) dynamics, and find excellent
correspondence. We discuss the implications of our results for stellar triples
and both stellar and supermassive triple black hole mergers.
