Gluon-gluon to photon-photon scattering gg→γγ offers to the LHC experiments a uniquely powerful probe of dimension-8 operators in the standard model effective field theory that are quadratic in both the electromagnetic and gluonic field-strength tensors, such as would appear in the Born-Infeld extension of the standard model. We use 13-TeV ATLAS data on the production of isolated photon pairs to set lower limits on the scales of dimension-8 operators M≳1 TeV and discuss the prospective sensitivities of possible future hadron colliders.