Theoretically, a regular expression can be expressed in constant space, and that's profoundly limiting, so anything that requires so much as a stack is not a regex.

That includes basic things like capture groups. That doesn't mean you can't use automata in a regex-plus implementation, though.

I can't find the paper, but I think there is a hybrid engine that tries to get the best of both worlds; constant space where possible and using additional space to provide more features.