Answer
In general, the cross product is not associative.
Work Step by Step
Let $\vec{A},\vec{B},\vec{C}$ be 3 non-zero vectors
Suppose $\vec{A} = \vec{B}\ne\vec{C}$ , then
$\vec{A}\times\vec{B}=0$ $\hspace{1cm}(\because$ they are along the same direction$)$
But, $\vec{B}\times\vec{C}\ne0$ and is perpendicular to both $\vec{B}$ and $\vec{C}\hspace{0.4cm}$ (By definiton of the cross product)
Thus, it is also perpendicular to $\vec{A}\hspace{0.4cm}$ ( $\because\vec{A}=\vec{B}$ )
$\therefore\vec{A}\times(\vec{B}\times\vec{C})$ is a non zero quantity, whereas
$(\vec{A}\times\vec{B})\times\vec{C}=0\hspace{0,4cm}(\because\vec{A}\times\vec{B}=0)$
$\therefore(\vec{A}\times\vec{B})\times\vec{C}\ne\vec{A}\times(\vec{B}\times\vec{C})$ ,i.e., the cross product between the vectors is not associative.
