  The estimation of repeatedly nested expectations is a challenging task that
arises in many real-world systems. However, existing methods generally suffer
from high computational costs when the number of nestings becomes large. Fix
any non-negative integer $D$ for the total number of nestings. Standard Monte
Carlo methods typically cost at least $\mathcal{O}(\varepsilon^{-(2+D)})$ and
sometimes $\mathcal{O}(\varepsilon^{-2(1+D)})$ to obtain an estimator up to
$\varepsilon$-error. More advanced methods, such as multilevel Monte Carlo,
currently only exist for $D = 1$. In this paper, we propose a novel Monte Carlo
estimator called $\mathsf{READ}$, which stands for "Recursive Estimator for
Arbitrary Depth.'' Our estimator has an optimal computational cost of
$\mathcal{O}(\varepsilon^{-2})$ for every fixed $D$ under suitable assumptions,
and a nearly optimal computational cost of $\mathcal{O}(\varepsilon^{-2(1 +
\delta)})$ for any $0 < \delta < \frac12$ under much more general assumptions.
Our estimator is also unbiased, which makes it easy to parallelize. The key
ingredients in our construction are an observation of the problem's recursive
structure and the recursive use of the randomized multilevel Monte Carlo
method.
