  We present some general results implying nonfinite axiomatisability of many
additively idempotent semirings with finitely based semigroup reducts. The
smallest is a $3$-element commutative example, which we show also has
\texttt{NP}-hard membership for its variety. As well as being the only
nonfinite axiomatisable ai-semiring on $3$-elements, we are able to show that
its nonfinite basis property infects many related semirings, including the
natural ai-semiring structure on the semigroup $B_2^1$. We also extend previous
group-theory based examples significantly, by showing that any finite
additively idempotent semiring with a nonabelian nilpotent subgroup is not
finitely axiomatisable for its identities.
