  It is by now well known that the Boltzmann-Gibbs (BG) entropy
$S_{BG}=-k\sum_{i=1}^W p_i \ln p_i$ can be usefully generalized into the
entropy $S_q=k (1-\sum_{i=1}^Wp_i^{q}) / (q-1)$ ($q\in \mathcal{R};
S_1=S_{BG}$). Microscopic dynamics determines, given classes of initial
conditions, the occupation of the accessible phase space (or of a
symmetry-determined nonzero-measure part of it), which in turn appears to
determine the entropic form to be used. This occupation might be a uniform one
(the usual {\it equal probability hypothesis} of BG statistical mechanics),
which corresponds to $q=1$; it might be a free-scale occupancy, which appears
to correspond to $q \ne 1$. Since occupancies of phase space more complex than
these are surely possible in both natural and artificial systems, the task of
further generalizing the entropy appears as a desirable one, and has in fact
been already undertaken in the literature. To illustrate the approach, we
introduce here a quite general entropy based on a distribution of $q$-indices
thus generalizing $S_q$. We establish some general mathematical properties for
the new entropic functional and explore some examples. We also exhibit a
procedure for finding, given any entropic functional, the $q$-indices
distribution that produces it. Finally, on the road to establishing a quite
general statistical mechanics, we briefly address possible generalized
constraints under which the present entropy could be extremized, in order to
produce canonical-ensemble-like stationary-state distributions for Hamiltonian
systems.
