  A search is presented for non-resonant Higgs boson pair production, targeting
the $bbZZ$, 4$V$ ($V$ = $W$ or $Z$), $VV\tau\tau$, 4$\tau$, $\gamma\gamma VV$
and $\gamma\gamma\tau\tau$ 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,
$\kappa_{\lambda}$, are determined to be $-6.2 < \kappa_{\lambda} < 11.6$
($-4.5 < \kappa_{\lambda} < 9.6$) at 95% confidence level, assuming the
Standard Model for the expected limits and that new physics would only affect
$\kappa_{\lambda}$.
