Large-deviation theory describes the statistics of rare fluctuations, while self-similarity captures scale-invariant behavior. Both are central pillars of statistical physics, yet the extent of their overlap is not well understood. Here we ask whether a stochastic process can simultaneously satisfy both. We show that, under a precise set of conditions — exact self-similarity governed by one Hurst exponent, and a scaled cumulant generating function that is analytic around the origin — compatibility holds only if the scaling distribution is Gaussian. We illustrate this result with a broad class of subordinated stochastic processes, including continuous-time random walks, diffusing-diffusivity models, and polymers in the grand-canonical ensemble. In non-Gaussian cases, a characteristic timescale separates regular large-deviation and self-similar regimes. At criticality, the divergence of the correlation time and the emergence of a singular cumulant generating function make a regular large-deviation description inappropriate and lead to a strong enhancement of extreme dynamical fluctuations.