13. $\frac{\pi}{2}$. Since the smallest positive period of $|\sin 2 x|$ is $\frac{\pi}{2}$, we have $\left|\sin 2\left(x+\frac{\pi}{2}\right)\right|=|\sin 2 x|$. For any real number $T$, when $0<T<\frac{\pi}{2}$, we have $|\sin 2(x+T)| \neq|\sin 2 x|$. Let $f(x)=3^{x}+2^{x}+5 x$, then $f(x)$ is a monotonically increasing function on $\mathbf{R}$. Therefore, $f\left[\left|\sin 2\left(x+\frac{\pi}{2}\right)\right|\right]=f(|\sin 2 x|)$, and for any real number $T$, when $0<T<\frac{\pi}{2}$, we have $f[|\sin 2(x+T)|] \neq f(|\sin 2 x|)$. Hence, the smallest positive period of $y=f(|\sin 2 x|)$ is $\frac{\pi}{2}$.
13. Find the smallest positive period of the function $f(x)=3 \sin 2 x+2^{\mid \sin ^{2} 2 x}+5|\sin 2 x|$.