A free algebra that is not generic

For example the free group [or free monoid] on one generator is surely not generic among all the groups [monoids], because it's commutative.

This question is somewhat related to this MO question. The MO question asks for examples of locally finite varieties that are not finitely generated. If a variety is locally finite, then its finitely generated free algebras are finite. If the variety is not finitely generated, then no finite algebra in the variety is generic. So the examples given at the link are of varieties where no free algebra of finite rank is generic.