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.