Trigonometry: Angle Sum and Difference Identities

If $-1 < x < 1$ and $-1 < y < 1$, define the "relativistic sum'' $x \oplus y$ to be $$x \oplus y = \frac{x + y}{1 + xy} .$$ The operation $\oplus$ is commutative and associative. Let $v$ be the number $$v = \frac{\sqrt[7]{17} - 1}{\sqrt[7]{17} + 1} .$$ What is the value of $$v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v?$$ (In this expression, $\oplus$ appears $13$ times.)

(Hint: What does $\frac{r-1}{r+1} \oplus \frac{s-1}{s+1}$ simplify to?)

Source: Math Prize for Girls