Simplify $\left(\frac{100}{101}\right)^{-3}\left(\frac{100}{101}\right)^{3}$.
By definition, if $a$ is nonzero, then $a^{-3}$ is the reciprocal of $a^3$.  So, $\left(\frac{100}{101}\right)^3$ and $\left(\frac{100}{101}\right)^{-3}$ are reciprocals.  Therefore, their product is $\boxed{1}$.