Nikolai Durov showed that a commutative algebraic monad with 0 "is" a semiring if and only if b(x,0)=x for all x, where b is a binary operation with b(x,y) not identically equal to x.
So semirings are in some sense easy to get.
On the other hand, the commutative algebraic monads that seem to be his motivating examples, the unit ball in a commutative Banach algebra, are not semirings.
