Example 2 The square of the sum of the digits of a natural number $k$ is denoted as $f_{1}(k)$, and $f_{n}(k)=f_{1}\left[f_{n-1}(k)\right]$, then the range of $f_{n}(11)(n \in$ $\left.\mathbf{N}^{*}\right)$ is
A. $\mathbf{N}^{*}$
B. 5
C. $\{4,16,49,169,256\}$
D. $\{2,4,7,13,16\}$
$$
\begin{array}{c}
\text { Solution: From the conditions, we know: } \\
f_{1}(11)=(1+1)^{2}=4 ; f_{2}(11)=f_{1}(4)=4^{2} \\
=16 ; \\
f_{3}(11)=f_{1}(16)=(1+6)^{2}=49 ; f_{4}(11)= \\
f_{1}(49)=(4+9)^{2}=169 ; \\
f_{5}(11)=f_{1}(169)=(1+6+9)^{2}=256 ; \\
f_{6}(11)=f_{1}(256)=(2+5+6)^{2}=169 ; \cdots
\end{array}
$$

Therefore, the range of $f_{n}(11)\left(n \in \mathbf{N}^{*}\right)$ is $\{4,16,49$, $169,256\}$.