This problem is studied in detail in <A HREF="https://doi.org/10.4169/math.mag.87.5.338">Iteration of Sine and Related Power Series</A>. Denoting the $n$-th iterate by $\sin^{\circ n}x$, one has the limit
$$\lim_{n\rightarrow\infty}\sqrt n\sin^{\circ n}(x/\sqrt n)=\frac{x}{\sqrt{1+x^2/3}}.$$

This is a nontrivial limit, without the rescaling $\sin^{\circ n}x\rightarrow 0$ (basically because $\frac{\sin x}{x}\leq 1$ and equality only for $x=0$, for a proof, see page 5 of <A HREF="https://arxiv.org/abs/1212.1027">arXiv:1212.1027</A>).