A search is presented for non-resonant Higgs boson pair production, targeting the bbZZ, 4V (V = W or Z), VVττ, 4τ, γγVV and γγττ decay channels. Events are categorised based on the multiplicity of light charged leptons (electrons or muons), hadronically decaying tau leptons, and photons. The search is based on a data sample of proton-proton collisions at $ \sqrt{s} $ = 13 TeV recorded with the ATLAS detector during Run 2 of the Large Hadron Collider, corresponding to an integrated luminosity of 140 fb$^{−1}$. No evidence of the signal is found and the observed (expected) upper limit on the cross-section for non-resonant Higgs boson pair production is determined to be 17 (11) times the Standard Model predicted cross-section at 95% confidence level under the background-only hypothesis. The observed (expected) constraints on the HHH coupling modifier, κ$_{λ}$, are determined to be −6.2 < κ$_{λ}$< 11.6 (−4.5 < κ$_{λ}$< 9.6) at 95% confidence level, assuming the Standard Model for the expected limits and that new physics would only affect κ$_{λ}$.[graphic not available: see fulltext]