  We extend the Hanel and Thurner asymptotic analysis to both extensive and
non-extensive entropies on the basis of a wide class of entropic forms. The
procedure is known to be capable to classify multiple entropy measures in terms
of their defining equivalence classes. Those are determined by a pair of
scaling exponents taking into account a large number of microstates as for the
thermodynamical limit. Yet, a generalisation to this formulation makes it
possible to establish an entropic connection between Markovian and
non-Markovian statistical systems through a set of fundamental entropies
$S_{\pm}$, which have been studied in other contexts and exhibit, among their
attributes, two interesting aspects: They behave as additive for a large number
of degrees of freedom while they are substantially non-additive for a small
number of them. Furthermore, an ample amount of special entropy measures,
either additive or non-additive, are contained in such asymptotic
classification. Under this scheme we analyse the equivalence classes of
Tsallis, Sharma-Mittal and R\'enyi entropies and study their features in the
thermodynamic limit as well as the correspondences among them.
