Answer
The cross product is not generally associative, as demonstrated with unit direction vectors $(\hat{i} \times \hat{i}) \times \hat{j}$ and $\hat{i} \times (\hat{i} \times \hat{j})$
However, in some special cases it can be. These cases include $(\hat{i} \times -\hat{j}) \times \hat{i}$ and $\hat{i} \times (-\hat{j} \times \hat{i})$
Work Step by Step
$(\vec{A} \times \vec{B}) \times \vec{C} \stackrel{?}{=}\vec{A} \times (\vec{B} \times \vec{C})$
Let $\vec{A} = \vec{B} = \hat{i}$ and $\vec{C} = \hat{j}$
$(\hat{i} \times \hat{i}) \times \hat{j} \stackrel{?}{=}\hat{i} \times (\hat{i} \times \hat{j})$
The cross product of parallel vectors is zero, and $\hat{i}\times \hat{j} = \hat{k}$
$0 \times \hat{j} \stackrel{?}{=} \hat{i}\times \hat{k}$
Anything crossed with zero is zero, and $\hat{i} \times \hat{k} = -\hat{j}$
$0\neq -\hat{j}$; the cross product is not associative
An example of three vectors that are associative with the cross product is $\hat{i}, -\hat{j}, \hat{i}$
$(\hat{i} \times -\hat{j}) \times \hat{i} \stackrel{?}{=}\hat{i} \times (-\hat{j} \times \hat{i})$
$-\hat{k} \times \hat{i} \stackrel{?}{=}\hat{i} \times \hat{k}$
$-\hat{j} = -\hat{j} \checkmark$
