On the Wikipedia page about Transcendental numbers there is a section about numbers that are proven to be transcendental and there you can read that transcendental numbers are sin(a), cos(a) and tan(a), and their multiplicative inverses csc(a), sec(a) and cot(a), for any nonzero algebraic number a (by the Lindemann–Weierstrass theorem).
Let $\sin_1(x)=\sin(x)$ and $\sin_2(x)=\sin(\sin(x))$, and, more generally, $\sin_n(x)=\sin_{n-1}(\sin(x))$ for every $n \in \mathbb N$.
Is it true that there exists nonzero algebraic number $\alpha$ such that the set $\{\sin_n(\alpha): n\in \mathbb N\}$ contains only transcendental numbers?