Non-set-theoretical foundations

Question

Nowadays most ideas of foundations are based on some set theories. But are there some non-set-theoretical foundations, I mean are there some ideas of creating a theory which can foundate other math theories and where fundamental concept is not a set but some other thing ? I know that Type theory and Category theory are kind of like that but are there some except these ? And I know there is a similar question Are there other approaches for the foundations of mathematics, other than logic and set theory?, but theories mentioned in answers to it are Category theory and Type theory. But are there some other examples ?