  In hierarchical triple systems, the inner binary is perturbed by a distant
companion. For large mutual inclinations, the Lidov-Kozai mechanism secularly
excites large eccentricity and inclination oscillations of the inner binary.
The maximal eccentricity attained, $e_{\rm max}$ is simply derived and widely
used. However, for mildly hierarchical systems (i.e. the companion is
relatively close and massive), non-secular perturbations affect the evolution.
Here we account for fast non-secular variations and find new analytic formula
for $e_{{\rm max}}$, in terms of the system's hierarchy level, correcting
previous work and reproducing the orbital flip criteria. We find that $e_{{\rm
max}}$ is generally enhanced, allowing closer encounters between the inner
binary components, thus significantly changing their interaction and its final
outcome. We then extend our approach to include additional relativistic and
tidal forces. Using our results, we show that the merger time of
gravitational-wave (GW) sources orbiting massive black-holes in galactic nuclei
is enhanced compared with previous analysis accounting only for the secular
regime. Consequently, this affects the distribution and rates of such GW
sources in the relevant mild-hierarchy regime. We test and confirm our
predictions with direct N-body and 2.5-level Post-Newtonian codes. Finally, we
calculate the formation and disruption rates of hot-Jupiters (HJ) in planetary
systems using a statistical approach, which incorporates our novel results for
$e_{{\rm max}}$. We find that more HJ migrate from further out, but they are
also tidally disrupted more frequently. Remarkably, the overall formation rate
of HJs remains similar to that found in previous studies. Nevertheless, the
different rates could manifest in different underlying distribution of observed
warm-Jupiters.
