jmc

We know that $|ab|=|a|\cdot |b|$. Therefore, $$\left|{\left(-2-2\sqrt{3}i\right)}^3\right|={\left|-2-2\sqrt{3}i\right|}^3$$Now, $$\left|-2-2\sqrt{3}i\right|=\sqrt{{\left(-2\right)}^2+{\left(-2\sqrt{3}\right)}^2}=4$$Therefore, our answer is $4^3=\boxed{64}$.