  For every positive integer $r$, we introduce two new cohomologies, that we
call $E_r$-Bott-Chern and $E_r$-Aeppli, on compact complex manifolds. When
$r=1$, they coincide with the usual Bott-Chern and Aeppli cohomologies, but
they are coarser, respectively finer, than these when $r\geq 2$. They provide
analogues in the Bott-Chern-Aeppli context of the $E_r$-cohomologies featuring
in the Fr\"olicher spectral sequence of the manifold. We apply these new
cohomologies in several ways to characterise the notion of
page-$(r-1)$-$\partial\bar\partial$-manifolds that we introduced very recently.
We also prove analogues of the Serre duality for these higher-page Bott-Chern
and Aeppli cohomologies and for the spaces featuring in the Fr\"olicher
spectral sequence. We obtain a further group of applications of our
cohomologies to the study of Hermitian-symplectic and strongly Gauduchon
metrics for which we show that they provide the natural cohomological
framework.
