Example 7 Given $f(f(f(x)))=8 x+7$, find the linear real-coefficient function $f(x)$.

 

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Let $f(x)=a x+b(a \neq 1, a, b \in \mathbf{R})$ be a linear function with real coefficients, then
$$
a^{3}\left(x-\frac{b}{1-a}\right)+\frac{b}{1-a}=8 x+7 \text {, }
$$

which implies $\left\{\begin{array}{l}a^{3}=8 \\ \frac{\left(a^{3}-1\right) b}{1-a}=-7\end{array} \Rightarrow\left\{\begin{array}{l}a=2, \\ b=1 .\end{array}\right.\right.$
Therefore, the required linear function with real coefficients is $f(x)=2 x+1$.