The ability to accurately calculate reaction barrier heights is of central importance to many areas of chemistry. We report an extensive study examining the basis set convergence of post-CCSD(T) contributions (up to CCSDT(Q)) for a diverse set of 28 reaction barrier heights. In contrast to previous studies, we focus here on larger transition structures (TSs) involving 4-7 non-hydrogen atoms. The set of reaction barrier heights includes pericyclic, bipolar cycloaddition, cycloreversion, and multiple-proton transfer reactions. We find that in most cases post-CCSD(T) contributions converge rapidly toward the basis set limit, such that even double-zeta and truncated double-zeta basis sets provide useful estimates of the T-(T) and (Q) contributions, respectively. In addition, we find that due to the tendency of these small basis sets to systematically underestimate the T-(T) and (Q) components, scaling is an effective approach for improving performance. For example, scaling the T-(T)/cc-pVDZ contribution by 1.25 results in an RMSD of merely 0.4 kJ mol(-1) relative to basis set limit reference values from W3lite-F12 theory. Similarly, calculating the (Q) contribution with a cc-pVDZ basis set without d functions and scaling by 1.6 results in an RMSD of 0.5 kJ mol(-1). We also examine the magnitude of post-CCSD(T) contributions for a wide range of TSs. We find that for pericyclic, bipolar cycloaddition, and multiple-proton transfer reactions there is an effective cancellation between the T-(T) and (Q) components (i.e., they have opposite signs and are of similar magnitude), such that overall post-CCSD(T) contributions to the reaction barrier heights are below similar to 1 kJ mol(1) (in absolute value). However, for the barrier heights of cycloreversion reactions, the T-(T) and (Q) components are both negative and large and consequentially post-CCSD(T) contributions reduce the reaction barrier heights by significant amounts ranging between 4.1 and 6.7 kJ mol(-1).