The observations of fully-charm tetraquark states in the LHCb, CMS and ATLAS experiments suggested the existence of the hadronic molecules of two-charmonium states, which may also imply bound states in the three-charmonium systems. In this work, we study the possible bound states in the triple-$\eta_c$ and triple-$J/\psi$ systems with $J^{PC}=0^{-+}$ and $1^{--}$, respectively. In QCD sum rules, we calculate the two-point correlation functions and spectral functions up to the dimension-four gluon condensate. We use the iterative dispersion relation approach to deal with the five-loop banana integrals, which significantly improves the computational efficiency. Our results show that the masses of triple-$\eta_c$ and triple-$J/\psi$ states lie below the corresponding mass thresholds, supporting the existence of such three-body bound states.
Comment: 7 pages, 5 figures. Accepted for publication in Chinese Physics Letters