  A search for nonresonant Higgs boson pair production in the
$b\bar{b}\gamma\gamma$ final state is performed using 140 fb$^{-1}$ of
proton-proton collisions at a centre-of-mass energy of 13 TeV recorded by the
ATLAS detector at the CERN Large Hadron Collider. This analysis supersedes and
expands upon the previous nonresonant ATLAS results in this final state based
on the same data sample. The analysis strategy is optimised to probe anomalous
values not only of the Higgs ($H$) boson self-coupling modifier
$\kappa_\lambda$ but also of the quartic $HHVV$ ($V=W,Z$) coupling modifier
$\kappa_{2V}$. No significant excess above the expected background from
Standard Model processes is observed. An observed upper limit $\mu_{HH}<4.0$ is
set at 95% confidence level on the Higgs boson pair production cross-section
normalised to its Standard Model prediction. The 95% confidence intervals for
the coupling modifiers are $-1.4<\kappa_\lambda<6.9$ and
$-0.5<\kappa_{2V}<2.7$, assuming all other Higgs boson couplings except the one
under study are fixed to the Standard Model predictions. The results are
interpreted in the Standard Model effective field theory and Higgs effective
field theory frameworks in terms of constraints on the couplings of anomalous
Higgs boson (self-)interactions.
