  We brief{}ly review the connection between statistical mechanics and
thermodynamics. We show that, in order to satisfy thermodynamics and its
Legendre transformation mathematical frame, the celebrated Boltzmann-Gibbs~(BG)
statistical mechanics is suff{}icient but not necessary. Indeed, the
$N\to\infty$ limit of statistical mechanics is expected to be consistent with
thermodynamics. For systems whose elements are generically independent or
quasi-independent in the sense of the theory of probabilities, it is well known
that the BG theory (based on the additive BG entropy) does satisfy this
expectation. However, in complete analogy, other thermostatistical theories
(\emph{e.g.}, $q$-statistics), based on nonadditive entropic functionals, also
satisfy the very same expectation. We illustrate this standpoint with systems
whose elements are strongly correlated in a specific manner, such that they
escape the BG realm.
