Below is a natural example of a ring object in the monoidal category of affine schemes. If $p$ is a prime and $R$ is a commutative ring, then the ring $W(R)$ of $p$-typical Witt vectors is also a commutative ring. The functor $W: CommRings \rightarrow CommRings$ is in fact co-representable by some commutative ring $\mathbb{W}$ which has two coproducts inducing the addition and multiplication, respectively, on $W(R)$ for each $R$. That is, the two coproducts give $hom_{CommRings}(\mathbb{W},R)$ a natural ring structure for all commutative rings $R$, and with that ring structure it is isomorphic to $W(R)$. Consequently, Spec of that co-representing ring is a ring object $Spec\ \mathbb{W}$ in the category of affine schemes. 

There is a nice, but unpublished, note about this affine ring scheme by Paul Pearson from the mid-2000s, which has a few calculations of the structure maps.