Does the series $\sum \sin^{(n)}(1)$ converge, where $\sin^{(n)}$ denotes the $n$-fold composition of $\sin$?

The series diverges. To see this, first note that $$ a_1 = 1\ge 1 $$ and that, if $a_n \ge 1/n$, then $$ a_{n+1} = \sin(a_n) \ge \sin(1/n) > 1/(n+1) $$ By induction, we have $a_n \ge 1/n$ for all $n$. Since $\sum\frac{1}{n}$ diverges, so does $\sum a_n$.

Note that $(n+1)\sin(1/n) > 1$ can be shown by Taylor expansion.