This paper gives a definition of biring and plethory, but here the latter are called biring triples. They make connections with Adams operations? and special lambda rings.
Theorem. A functor $\mathsf{CRing} \to \mathsf{CRing}$ is representable if and only iff it has a left adjoint.
The identity functor on $\mathsf{CRing}$ is represented by the ring $\mathbb{Z}[x]$.
The functor which sends a ring to its own power series ring is represented by $\mathbb{Z} [X_0, X_1, \dots]$.