Let $\mathbb{A}_1$ be the set of algebraic numbers. The usual definitions of this set is the roots of all non-trivial polynomials with integer coefficients.
The set of transcendental numbers is defined as the non-algebraic reals. However it seems reasonable to define the set $\mathbb{A}_n$ as the set of solutions of all non-trivial polynomials with coefficients in $\mathbb{A}_{n-1}$.
Is this something looked into? If yes is the set $\mathbb{A}_\infty$ equal to $\mathbb{R}$? In which $\mathbb{A}_n$ are the famous transcendental numbers as $\pi$, $\phi$ and e?
(I am not sure about the tags I should use. So feel free to change them.)
Any root of a polynomial equation with algebraic coefficients is still algebraic. In other words, the field of algebraic numbers is algebraically closed. Therefore, $\mathbb{A}_1=\mathbb{A}_2=\mathbb{A}_3=\cdots$