The notion of nilpotency passes nicely from groups to semigroups. Define $q_1(x,y)=xy$ and $$q_{i+1}(x,y,z_1,\cdots,z_i)=q_i(x,y,z_1,\cdots,z_{i-1})z_iq_i(y,x,z_1,\cdots,z_{i-1})$$ inductively for all $x,y,z_1,z_2,\cdots$.

A. I. Malcev proved that nilpotent groups of class $c$ are described as the groups satisfying the law $q_c(x,y,z_1,\ldots,z_{c-1})=q_c(y,x,z_1,\ldots,z_{c-1})$. It is natural to define nilpotent semigroups of class $c$ as those satisfying this law.

Question: Do all f.g. nilpotent semigroups have polynomial growth?

Note that for cancellative semigroups the answer is "yes" since nilpotent cancellative semigroups satisfy the Ore's condition and so are group-embeddable.

(My guess is that in general the answer is "no" and there even perhaps exists a counter-example among matrix semigroups.)