The set $\{0\}$ is a ring with identity.

Reason for exclusion: it is an edge case that does not mix well with the deduction software infrastructure.

The surreal numbers satisfy all axioms of an ordered field except that the elements of the surreal numbers are not a set, but a proper class. All other ordered fields appear within the surreal numbers.

Knuth, Donald Ervin. "Surreal numbers: how two ex-students turned on to pure mathematics and found total happiness: a mathematical novelette." (1974).

Reason for exclusion: the underlying collection does not form a set: it is a proper class.

Smoktunowicz constructed a rng which has only trivial ideals and for which every element is nilpotent.

Smoktunowicz, Agata. "A simple nil ring exists." Communications in Algebra 30.1 (2002): 27-59.

Reason for exclusion: no identity.