My problem:
Let $\alpha$ be a closed path of $[0,1]$ in the union of three no disjoint open balls $B_1,B_2,B_3\subset\mathbb{R}^n$ such that $B_1\cap B_2\cap B_3=\emptyset$, if the base point of $\alpha$ is $x_0\in B_1\cap B_2$, then $\alpha$ through only finitely many times the intersection $B_2\cap B_3$?
Note that through finitely many times $B_2\cap B_3$ means that $\alpha^{-1}(B_2\cap B_3)$ is finite union of disjoint open subintervals of [0,1].
I need this result to prove that the fundamental group of this three balls are finitely generate by lines connecting fixed points in his intersections.
No. $\alpha$ can go only infinitely many times through the intersection $B_2 \cap B_3$. Take a path from $x_0$ to a boundary point $x_1 \in \text{bd} B_3 \cap B_2$. Then you can run in finite time through infinitely loops $l_n$ in $\overline B_3 \cap B_2$ of length $2^{-n}$ such that $l_n \cap \text{bd} B_3 = \{x_1\}$. Finally take any path from $x_1$ to $x_0$. Drawing a picture is helpful.