= Solution
For fixed real $x$, define $u_0=x$ and $u_n=\sin(0.99u_{n-1})$. This makes $u_n$ exactly the summand containing $n$ nested sines. The elementary inequality $|\sin t|\le|t|$ gives inductively
$$
|u_n|\le0.99|u_{n-1}|\le0.99^n|x|.
$$
The fact that an <iterated contraction gives a summable orbit> applies here. Comparison with a <geometric series> therefore gives
$$
\sum_{n\ge1}|u_n|\le |x|\frac{0.99}{1-0.99}=99|x|<\infty.
$$
Thus \b[the nested-sine <series> converges absolutely for every real $x$; there are no divergent or merely conditionally convergent cases].
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