1. Given the function
$$
f(x)=\arcsin (\cos x) \text {. }
$$

then the smallest positive period of $f(f(f(x)))$ is $\qquad$ .
$-1 . \pi$.
From the fact that $\cos x$ is an even function, we know that $f(x)$ is an even function.
$$
\begin{array}{l}
\text { and } f(x+\pi)=\arcsin [\cos (x+\pi)] \\
=\arcsin (-\cos x)=-\arcsin (\cos x) \\
=-f(x),
\end{array}
$$

Therefore, $f(f(x+\pi))=f(-f(x))=f(f(x))$. Thus, the period of $f(f(x))$ is $\pi$.
Hence, the smallest positive period of $f(f(f(x)))$ is $\pi$.