Let $\def\A{\mathfrak A}\A$ be the smallest subset of $\Bbb C$ that contains the algebraic numbers and also all numbers of the form $$\sum \alpha_i^{\beta_i}$$ where the $\alpha_i, \beta_i$ are themselves elements of $\A$. This class $\A$ is much larger than the algebraic numbers. For example, it contains $2^{\sqrt 2}$, which is known to be transcendental, by the Gel'fond–Schneider theorem.
Does $\A$ have a name? Is anything known about $\A$? In particular, is it known whether $e$ and $\pi$ are in $\A$?