Real numbers that are not solutions of polynomial equations with integer coefficients are called transcendental numbers.
Now consider the set of functions $S = \left\{\sin(x), 1, x, x^2, x^3, ... , x^n, ...\right\}$. Let $\bar{S}$ be the closure of $S$ under function composition and linear combinations with integer coefficients. For example $x^7 + 27x^5 + 3(\sin(x^3))^3 + \sin(\sin(x + \sin(2x))) + 1 \in \bar{S}$.
What numbers are the roots of such equations? For example $\pi$ is a root of $\sin(x) = 0$ so it is no longer transcendental. Is the set of roots countable like in the case of algebraic numbers? What if we add other functions to S, for example we add all elementary functions to S? What if we also allow multiplication of functions, in order to make S a field?