Let A denote the class of aperiodic monoids with central idempotents. A subvariety of A that is not contained in any finitely generated subvariety of A is said to be inherently non-finitely generated. A characterization of inherently non-finitely generated subvarieties of A, based on identities that they cannot satisfy and monoids that they must contain, is given. It turns out that there exists a unique minimal inherently non-finitely generated subvariety of A, the inclusion of which is both necessary and sufficient for a subvariety of A to be inherently non-finitely generated. Further, it is decidable in polynomial time if a finite set of identities defines an inherently non-finitely generated subvariety of A.