A rescaling is needed for a nontrivial limit. As discussed in <A HREF="https://www.jstor.org/stable/10.4169/math.mag.87.5.338#metadata_info_tab_contents">Iteration of Sine and Related Power Series</A>, denoting the $n$-th iterate by $\sin^{\circ n}x$, one has the limit
$$\lim_{n\rightarrow\infty}\sqrt n\sin^{\circ n}(x/\sqrt n)=\frac{x}{\sqrt{1+x^2/3}}.$$
The graph (from the cited paper) shows that the limit is attained quite rapidly.
<IMG SRC="https://ilorentz.org/beenakker/MO/sineiterate_1.png"/>

Without the rescaling the iterated sine converges to zero (for the reasons indicated in the comments to the OP). The convergence is slow, see the graph.

<IMG SRC="https://ilorentz.org/beenakker/MO/sineiterate_2.png"/>