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There is a nice, but unpublished, note about this affine ring scheme by Paul Pearson from the mid-2000s, which has a few calculations of the structure maps.
Another class of examples is provided by Hopf rings, which are ring objects in the category of cocommutative coalgebras. These things arise naturally in topology from studying unstable operations on generalized homology theories. The point is that, if you have a suitably nice spectrum $E$, then when you evaluate $E$ on the spaces in the $\Omega$-spectrum for $E$ and take the direct sum over all those spaces in the $\Omega$-spectrum, the resulting bi-graded gadget inherits a lot of structure from the structure of $E$: in fact, it picks up the structure of a Hopf ring. This is pretty well-documented in various resources on the Web, e.g. here: https://encyclopediaofmath.org/wiki/Hopf_ring