Part of why I want this example, is because I want to see if the characteristic of a ring must be an element of the ring itself.
It is not really sensible to think of the characteristic of a ring as being an element of the ring, even when it literally is.
Characteristic is a sort of "dimension" on the ring, having a value in the natural numbers.
It's kind of like asking if the dimension of a vector space can be an element in the vector space or not.
Now, coincidentally the characteristic of $\mathbb Z$ is in $\mathbb Z$ if you like $\mathbb N$ being a subset of $\mathbb Z$, but the observation is not really useful.
The closest you can come to putting the characteristic "in" the ring is if the ring has identity so that you can see the homomorphic image of $\mathbb Z$ within your ring. But even then, the characteristic always maps to the $0$ element of your ring, so nothing is achieved.
An example, though?
I dunno, can you think of an abelian group $G$ that doesn't involve numbers? if so, the endomorphism ring $End(G_\mathbb Z)$ is a ring that apparently does not involve numbers.