A rescaling is needed for a nontrivial limit. As discussed in Iteration of Sine and Related Power Series by C. Towse (2014), 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.
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.
For the general rescaling, $$z_\alpha(x)=\lim_{n\rightarrow\infty}n^\alpha\sin^{\circ n}(n^{-\alpha}x),$$ I surmise (based on the small-$x$ expansion of the sine$^\ast$) that the limit is $z_\alpha(x)=0$ for $\alpha<1/2$ and $z_\alpha(x)=x$ for $\alpha>1/2$. This agrees quite well with the numerics, see plots below
Plots of $n^\alpha\sin^{\circ n}(n^{-\alpha}x)$ for $n=500$ and $\alpha=0.75$ (left) and $\alpha=0.25$ (right). The horizontal lines in the right plot show the asymptote $(\text{sign}\,x)\,\sqrt{3}\,n^{\alpha-1/2}$.
$^\ast$ For $\alpha<1/2$ we can proceed as follows: Assume a power law decay, $y_n=(\text{sign}\,x)\,n^\alpha\sin^{\circ n}(n^{-\alpha}x)=(\text{sign}\,x)cn^{-p}$, substitute into $y_{n+1}=(n+1)^\alpha \sin(y_n/n^\alpha)$, and expand for large $n$. So we have $$c(n+1)^{-p}\approx cn^{-p}-cpn^{-p-1},\;\; (n+1)^\alpha\sin(cn^{-p-\alpha})\approx cn^{-p}+\alpha cn^{-p-1}-\tfrac{1}{6}c^3n^{-3p-2\alpha},$$ and equating these two expressions gives $p=\tfrac{1}{2}-\alpha$, $c=\sqrt{6(p+\alpha)}=\sqrt{3}$. We thus arrive at the large-$n$ asymptotics $$n^\alpha\sin^{\circ n}(n^{-\alpha}x)\rightarrow (\text{sign}\,x)\,\sqrt{3}\,n^{\alpha-1/2}\rightarrow 0,\;\;\text{for}\;\;\alpha<1/2.$$
