jmc

Note that $f(x)=\sqrt[3]{20x+\sqrt[3]{20x+13}}$ is an increasing function, so the solution to $$\sqrt[3]{20x+\sqrt[3]{20x+13}}=13$$is unique. Furthermore, if $\sqrt[3]{20x+13}=13,$ then $x$ satisfies the given equation. Thus, $20x+13=13^3=2197,$ so $x=\boxed{\frac{546}{5}}.$