Answer
The cross product is not associative.
Work Step by Step
Let $\vec{a},\vec{b}$ be two nonzero orthogonal vectors.
If the cross product is associative, the following must be true:
$|(\vec{a} \times \vec{a}) \times \vec{b}| = |\vec{a} \times (\vec{a} \times \vec{b})|$
For the left side:
$\vec{a}\times \vec{a} = 0$ (it is the cross of two parralel vectors)
$|0 \times \vec{b}| = 0$
For the right side:
$\vec{a}\times \vec{b} = \vec{c}$, where $|\vec{c}| = ab$ and $\vec{a} \perp \vec{c}$
$|\vec{a} \times \vec{c}| = |a||ab|sin(\frac{\pi}{2}) = a^2b$
$a$ and $b$ are both nonzero, so $a^2b \neq 0$
Our premise is false, and the cross product is not associative.
