Light axion-like particles (ALPs) that couple to photons can mediate light-by-light scattering; in classical field theory terms, they induce cubic nonlinearities in Maxwell's equations. I will describe a new program of experiments to detect ALP-mediated light-by-light scattering which will also be sensitive to the Euler-Heisenberg Lagrangian, the light-by-light scattering process resulting from integrating out the electron in QED. This purely Standard Model effect has never been directly observed at energies below the electron mass. As the proposed experiment becomes more sensitive, the Euler-Heisenberg Lagrangian may seem to be an irreducible background to ALP searches, but may in fact be tuned away with a suitable choice of electric and magnetic fields in superconducting cavities. The optimistic endpoint of this program may be sensitive to an individual ALP with coupling to photons of 10^-12 GeV^-1, or equivalently, an "axiverse" of 10^8 light ALPs with decay constants at the renormalized Planck scale. I will briefly comment on the relevance of this experimental program for constraining the landscape of string compactifications.