9. From the problem, for any $n, f(n)=k$ is a function defined on the set of non-negative integers, taking values in $\{0,1,2,3,4,5,6,7,8,9\}$.
Let $f_{(2)}(n)=f(f(n)), f_{(3)}(n)=f\{f(f(n))\}, f_{(k)}(n)$ denote the $k$-fold composition of $f$.
Then $f_{(2)}(n)$ is a function defined on $\{0,1,2,3,4,5,6,7,8,9\}$, taking values in $\{0,1,2, \cdots, 9\}$.
The values can be listed as follows:
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline$f_{(1)}(n)$ & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\
\hline$f_{(3)}$ & 3 & 1 & 4 & 1 & 5 & 9 & 2 & 6 & 5 & 3 \\
\hline$f_{(3)}$ & 1 & 1 & 5 & 1 & 9 & 3 & 4 & 2 & 9 & 1 \\
\hline$f_{(n)}$ & 1 & 1 & 9 & 1 & 3 & 1 & 5 & 4 & 3 & 1 \\
\hline$f_{(3)}$ & 1 & 1 & 3 & 1 & 1 & 1 & 9 & 5 & 1 & 1 \\
\hline$f_{(n)}$ & 1 & 1 & 1 & 1 & 1 & 1 & 3 & 9 & 1 & 1 \\
\hline$f_{(3)}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 3 & 1 & 1 \\
\hline$f_{(n)}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline$f_{(n)}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline$f_{(n)}$ & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\
\hline
\end{tabular}

Thus, for any non-negative integer $n$:
$$
F(n)=\underbrace{\{}_{\text {fin } f\{f\{f \cdots f}\{f(n)\} \cdots\}\}\}=1
$$

Since $f(1990)+f(5)+f(13)$ is a non-negative integer, we have:
$$
F[f(1990)+f(5)+f(13)] \equiv 1 .
$$
9. (Adapted from the 1st "Hope Cup" Senior High School Competition) Let the function $f(n)=k$, where $n$ is a natural number, and $k$ is the digit at the $n$-th position after the decimal point of the irrational number $\pi=3.1415926535 \cdots$, with the rule that $f(0)=3$. Let $F_{n}=$ $\underbrace{f\{f\{f\{f\{f}(n)\} \cdots\}\}$, then $F[f(1990)+f(5)+f(13)]=$ $\qquad$.