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
$$\lim_{n\rightarrow\infty}\sqrt n\sin^{\circ n}(x/\sqrt n)=\frac{x}{\sqrt{1+x^2/3}}.$$