This problem is studied in detail in Iteration of Sine and Related Power Series. 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 $x=0$).