  Benchmark calculations on noncovalent interactions typically exclude
correlation effects beyond valence CCSD(T) owing to their steep computational
cost scaling. In this work, we consider their importance for water clusters,
specifically, eight isomers of (H$_2$O)$_6$ and four Wales-Hodges isomers of
(H$_2$O)$_{20}$. Higher order connected triples, $T_3$--(T), reduce
dissociation energies of the latter by about 0.4 kcal/mol, but this is more
than compensated by an increase of up to 0.85 kcal/mol due to connected
quadruple excitations. In general, higher-order correlation effects favor more
compact isomers over more `spread-out' ones. We also consider additional small
effects for balance: scalar relativistics reduce binding in (H$_2$O)$_{20}$ by
ca. --0.4 kcal/mol, which fortuitously is compensated by the ca. 0.55 kcal/mol
diagonal Born-Oppenheimer correction. Core-valence correlation has the greatest
impact, at ca. 1.3 kcal/mol for the icosamer.
