I'm interested in playing with nonwell-founded variants of set theory and weaker/different axioms of induction/extensionality.
I have a hunch coalgebraic methods could better handle weirdness like modelling homotopy type theory.
I also have just been interested in the idea of coinduction as primitive as opposed to induction.
Weak limits like weak function spaces are also useful in Computer Science for higher order abstract syntax. So set-theory minus induction/extensionality could lead to cleaner encodings of stuff like the lambda calculus.
But I really don't know where to start with nonwell-founded set theory in the first place. Unfortunately nonwell-founded set theory seems a bit esoteric.