$\max(\sin(\sin^2 x+2\sin x))=?$

Let $y = \sin \left(\sin^2 x + 2\sin x\right) = \sin \left((\sin x + 1)^2 - 1\right)$ . This implies that $\max(y) = \sin \left(\frac \pi 2\right) = \boxed 1$ , when $(\sin x +1)^2 - 1 = \frac \pi 2$ $\implies \sin x = \sqrt{\frac \pi 2+1} - 1 \approx 0.603$ , which exists.