We show experimentally and theoretically that the persistence of large deviations in equilibrium step fluctuations is characterized by an infinite family of independent exponents. These exponents are obtained by carefully analyzing dynamical experimental images of Al/Si(111) and Ag(111) equilibrium steps fluctuating at high (970 K) and low (320 K) temperatures, respectively, and by quantitatively interpreting our observations on the basis of the corresponding coarse-grained discrete and continuum theoretical models for thermal surface step fluctuations under attachment/detachment (″high-temperature″) and edge-diffusion limited kinetics (″low-temperature″), respectively.