The rational numbers are closed under addition and multiplication, with inverses for both. But what happens if you introduce exponentiation and ask for closure under it? I know there isn't a meaningful inverse, and that you immediately expand into the irrationals and complex numbers, but do you get all the complex numbers? If yes, how can you show this? If not, has this subset of the complex plane been studied?
Edit 1: I don't mean only taking rational numbers to rational number powers, I mean you start with rationals to rational powers, but then can use any resulting numbers as base or exponent.
Edit 2: Upon reflection I am pretty sure that because you start with a countably infinite set, and you only insist on closure under an operation applied at most a finite number of times, the resulting set will only be countably infinite. So it cannot cover the complex plane.
Edit 3: Karl pointed out that exponentiation isn't always single-valued, so in any case where there multiple solutions, we could insist that all are contained in the closure set.