Example 5 Given the function $f(x)=\frac{4}{5}(x-1)$. If $f^{(5)}(x)$ is a positive integer, find the minimum value of $x$.

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Let $\frac{4}{5}(x-1)=x$, solving for $x$ gives $x=-4$. Then $f(x)=\frac{4}{5}(x-1)=\frac{4}{5}(x+4)-4$.
So, $f^{(2)}(x)=\left(\frac{4}{5}\right)^{2}(x+4)-4$,
$$
f^{(5)}(x)=\left(\frac{4}{5}\right)^{5}(x+4)-4 \text {. }
$$

Since $f^{(5)}(x)$ is a positive integer, then $5^{5} \mid (x+4)$. Therefore, the minimum value of $x$ is $5^{5}-4=3121$.