The Jacobi-Anger expansion gives expressions for functional iterates of trigonometric functions in terms of Bessel functions. For instance, we have $$\sin(z \sin(\theta)) = 2 \sum_{n=1}^{\infty} J_{2n-1} (z) \sin[(2n-1)\theta] .$$ If we plug in $z=1$, we obtain an expression for $\sin(\sin(\theta)) := \sin^{[2]}(\theta)$. We might call this the second functional iterate of the sine function.
Are Jacobi-Anger expansions for $\sin^{[k]}(\theta)$ also known, for arbitrary $k>2$?