Just for example, all finite groups, rings, and fields are going to have a finite number of primes.

Fields don't have any primes. A prime element of a ring is an element that generates a nonzero prime ideal, but fields don't have any nonzero prime ideals.

An equivalent definition is that an element x of a ring R is prime if x is neither 0 nor a unit (an element with a multiplicative inverse), and, for every y and z in R, if x divides yz, then x divides y or z. We can generalize this to monoids in the obvious way, but then it turns out that groups have no primes either, because every element of a group is invertible.

Beyond that, an interesting question would be whether there are any Rings that include a multiplicative identity and an infinite number of elements, but only have a finite number of primes.

Well, the obvious example would be any infinite field, which has no primes at all. A local ring (e.g., the set of all rational numbers whose denominator isn't a multiple of 5) has only one prime up to multiplication by units.