  We study dynamical notions lying between $\mathcal{U}$-frequent
hypercyclicity and reiterative hypercyclicity by investigating weighted upper
densities between the unweighted upper density and the upper Banach density.
While chaos implies reiterative hypercyclicity, we show that chaos does not
imply $\mathcal{U}$-frequent hypercyclicity with respect to any weighted upper
density. Moreover, we show that if $T$ is $\mathcal{U}$-frequently hypercyclic
(resp. reiteratively hypercyclic) then the n-fold product of $T$ is still
$\mathcal{U}$-frequently hypercyclic (resp. reiteratively hypercyclic) and that
this implication is also satisfied for each of the considered
$\mathcal{U}$-frequent hypercyclicity notions.
