Three-body dispersion interactions are much weaker than their two-body counterpart. However, their importance grows quickly as the number of interacting monomers rises. To explore the numerical performance of correlation methods for long-range three-body dispersion, we performed calculations on eight very simple dispersion-dominated model metal trimers: Na3, Mg3, Zn3, Cd3, Hg3, Cu3, Ag3, and Au3. One encouraging aspect is that relatively small basis sets of augmented triple-ζ size appear to be adequate for three-body dispersion in the long-range. Coupled cluster calculations were performed at high levels to assess MP3, CCSD, CCSD(T), empirical density functional theory dispersion (D3), and the many-body dispersion (MBD) approach. We found that the accuracy of CCSD(T) was generally significantly lower than for two-body interactions, with errors sometimes reaching 20% in the investigated systems, while CCSD and particularly MP3 were generally more erratic. MBD is found to perform better than D3 at large distances, whereas the opposite is true at shorter distances. When computing reference numbers for three-body dispersion, care should be taken to appropriately represent the effect of the connected triple excitations, which are significant in most cases and incompletely approximated by CCSD(T).