  A landmark result from rational approximation theory states that $x^{1/p}$ on
$[0,1]$ can be approximated by a type-$(n,n)$ rational function with
root-exponential accuracy. Motivated by the recursive optimality property of
Zolotarev functions (for the square root and sign functions), we investigate
approximating $x^{1/p}$ by composite rational functions of the form $r_k(x,
r_{k-1}(x, r_{k-2}( \cdots (x,r_1(x,1)) )))$. While this class of rational
functions ceases to contain the minimax (best) approximant for $p\geq 3$, we
show that it achieves approximately $p$th-root exponential convergence with
respect to the degree. Moreover, crucially, the convergence is doubly
exponential with respect to the number of degrees of freedom, suggesting that
composite rational functions are able to approximate $x^{1/p}$ and related
functions (such as $|x|$ and the sector function) with exceptional efficiency.
