We study the discovery potential of axion-like particles (ALP), pseudo-scalars weakly coupled to Standard Model fields, at the Large Hadron Collider (LHC). Our focus is on ALPs coupled to the electromagnetic field, which would induce anomalous scattering of light-by-light. This can be directly probed in central exclusive production of photon pairs in ultra-peripheral collisions at the LHC in proton and heavy-ion collisions. We consider non-standard collision modes of the LHC, such as argon-argon collisions at sNN=7 TeV and proton-lead collisions at sNN=8.16 TeV, to access regions in the parameter space complementary to the ones previously considered for lead-lead and proton-proton collisions. In addition, we show that, using laser beam interactions, we can constrain ALPs as resonant deviations in the refractive index induced by anomalous light-by-light scattering effects. If we combine the aforementioned approaches, ALPs can be probed in a wide range of masses from the eV scale up to the TeV scale.