The understanding of the shape of the Higgs potential requires experimental information on the Higgs boson self-couplings. While current experiments already provide some insights on the trilinear self-coupling κ3 through Higgs pair production, important information will also become accessible from the production process of three Higgs bosons in the future runs of the LHC. Even though this process suffers from small cross sections, it is of particular interest since it depends on both κ3 and the quartic self-coupling κ4. Using Graph Neural Networks we investigate the feasibility of constraining the Higgs self-couplings through HHH production and find that the HL-LHC can be sensitive to large deviations from the SM that are allowed by the constraints from tree-level perturbative unitarity and in an effective field theory treatment are compatible with the present bounds on κ3. We additionally address the prospects of lepton colliders operating at the TeV energy range.