Basis set convergence of correlation effects on molecular atomization energies beyond the coupled cluster with singles and doubles (CCSD) approximation has been studied near the one-particle basis set limit. Quasiperturbative connected triple excitations, (T), converge more rapidly than L −3 L−3 (where L L is the highest angular momentum represented in the basis set), while higher-order connected triples, T 3 −(T) T3−(T) , converge more slowly—empirically, ∝L −5∕2 ∝L−5∕2 . Quasiperturbative connected quadruple excitations, (Q), converge smoothly as ∝L −3 ∝L−3 starting with the cc-pVTZ basis set, while the cc-pVDZ basis set causes overshooting of the contribution in highly polar systems. Higher-order connected quadruples display only weak, but somewhat erratic, basis set dependence. Connected quintuple excitations converge very rapidly with the basis set, to the point where even an unpolarized double-zeta basis set yields useful numbers. In cases where fully iterative coupled cluster up to connected quintuples (CCSDTQ5) calculations are not an option, CCSDTQ(5) (i.e., coupled cluster up to connected quadruples plus a quasiperturbative connected quintuples correction) cannot be relied upon in the presence of significant nondynamical correlation, whereas CCSDTQ(5) Λ CCSDTQ(5)Λ represents a viable alternative. Connected quadruples corrections to the core-valence contribution are thermochemically significant in some systems. We propose an additional variant of W4 theory [A. Karton et al., J. Chem. Phys.125, 144108 (2006)], denoted W4.4 theory, which is shown to yield a rms deviation from experimental atomization energies (active thermochemical tables, ATcT) of only 0.05kcal∕mol 0.05kcal∕mol for systems for which ATcT values are available. We conclude that “3σ⩽1kJ∕mol 3σ⩽1kJ∕mol ” thermochemistry is feasible with current technology, but that the more ambitious goal of ±10cm −1 ±10cm−1 accuracy is illusory, at least for atomization energies.