In ultra-relativistic heavy-ion collisions, large rates of $\gamma\gamma$ processes occur through the interaction of the large electromagnetic fields of the nuclei. These interactions enable the study of processes potentially sensitive to physics beyond the Standard Model. In ultra-peripheral collisions, characterized by large impact parameter between the nuclei, the outgoing particles exhibit back-to-back production in the transverse plane, which provides precise and efficient identification. This document presents an overview of recent ATLAS measurements potentially sensitive to physics beyond the Standard Model, including the production of tau leptons, and the production of magnetic monopoles. Measurements of tau lepton production help to constrain its anomalous magnetic moment, a quantity potentially sensitive to physics beyond the Standard Model. Also presented is a more recent search for monopole-pair production in ultra-peripheral collisions, with monopole masses ranging from 20 GeV to 150 GeV. The results are compared with recently developed semi-classical models that include non-perturbative cross section calculations.