  We carry out a study in the approach of a dimension-8 effective field theory
on the anomalous quartic $W^+W^-\gamma\gamma$ couplings that arise from the
process $pp \to W\gamma\gamma$ at the Future Circular Collider-hadron hadron
(FCC-hh). The process is sensitive to the $W^+W^-\gamma\gamma$ vertex and
limits may be placed on the anomalous $f_ {M,i}/\Lambda^4$ and $ f_
{T,j}/\Lambda^4$ couplings by measurements of $W\gamma\gamma$ production. We
illustrate the physics opportunities and reach of the FCC-hh operating for a
high integrated luminosity scenario of ${\cal L}=1, 5, 10, 30$ ${\rm ab}^{-1}$
of proton-proton collisions at a center-of-mass energy of $\sqrt{s}=100$ TeV.
Production cross-sections of the signal and the sensitivity constraints on the
anomalous $f_ {M,i}/\Lambda^4$ and $ f_ {T,j}/\Lambda^4$ couplings at $95\%$
C.L. are identified through the leptonic and hadronic decay modes of the
$W$-boson. The best FCC-hh sensitivity limits on these anomalous couplings
might reach up to the order of magnitude ${\cal O}(10^{-4}-10^1)$. These
results show that the $pp \to W\gamma\gamma$ channel at the FCC-hh has a
greater potential to research for the anomalous quartic $W^+W^-\gamma\gamma$
couplings and to research for new physics.
