Let $S$ denote the smallest subset of $\mathbb{R}_{>0}$ which includes $1$, and is closed under addition, multiplication, reciprocation, and the function $x,y \in \mathbb{R}_{>0} \mapsto x^y.$
Questions:
1. Do either $S$ or its elements have an accepted name?
2. Where can I learn more about the set $S$ and it elements? (Reference Request)
3. Do there exist positive, real algebraic numbers which are not in $S$?
4. Are either $e$ or $\pi$ elements of $S$?
What I Already Know:
By the Gelfond-Schneider Theorem, $S$ includes some transcendental numbers, like $2^{\sqrt{2}}$.