Assuming that N grows linearly with system size, the asymptotic cost of MBE(n) then grows as O(N n ), absent some kind of screening approximation to exclude unimportant subsystems. The combinatorics associated with Eq. ( ) can lead to catastrophic loss-ofprecision, e.g., when embedding charges are used, or for n > 4. High-order expansions with n ≥ 5 have seen little use, and mostly in small molecular clusters. Very few studies have examined higherorder terms for N ≥ 20 fragments. Much of the work with high-order MBE(n) methods has focused on the effects of basis-set superposition error (BSSE), which affects the n-body subsystems differently from how it affects the supersystem, leading to artifacts when the supersystem result is used to assess convergence. To address this, many-body versions of the Boys-Bernardi counterpoise correction have been developed. By means of these corrections, or else simply by using very large basis sets to eliminate BSSE, it has been determined that MBE(n) for neat water clusters converges at n = 4, or possibly n = 5. Higher-order n-body interactions persist in ionwater clusters and in some response properties, whereas MBE(n) for proteins appears to converge at n = 3, using single-residue fragments. These observations suggest that the vast majority of n-body interactions are insignificant so long as BSSE is taken into account.