Example 7 As shown in Figure 3.1.7, D is the incenter of $\triangle A B C$, E is the incenter of $\triangle A B D$, F is the incenter of $\triangle B D E$. If the degree of $\angle B F E$ is an integer, find the minimum degree of $\angle B F E$.
Solve: By the property of the incenter, we know
$$
\begin{aligned}
\angle B F E= & 90^{\circ}+\frac{1}{2} \angle B D E=90^{\circ}+\frac{1}{4} \angle B D A= \\
& 90^{\circ}+\frac{1}{4}\left(90^{\circ}+\frac{1}{2} \angle A C B\right)= \\
& 112^{\circ}+\frac{1}{8}\left(4^{\circ}+\angle A C B\right)
\end{aligned}
$$

Therefore, when $\angle A C B=4^{\circ}$, the minimum degree of $\angle B F E$ is $113^{\circ}$.