Definition. A set is an ordinal if it is a transitive set of transitive sets.
This is the simplest definition of an ordinal I have ever encountered, and I happen to like it a lot for this reason. However, it seems to have one downfall: I am having a lot of trouble showing that the class of ordinals is well-ordered without using the axiom of regularity (foundation). Do I need to use regularity to show this? Are there models of $ZFC-(\text{regularity})$ in which the transitive sets of transitive sets are not well-founded?
If you remove the axiom of foundation from ZF, then you cannot exclude the existence of, for instance, sets satisfying $x = \{x\}$. Such a set would be transitive, and have only transitive sets as elements, so it would fall under your definition of an ordinal.