Minimal massive gravity (MMG) is an extension of three-dimensional topologically massive gravity that, when formulated about anti-de Sitter space, accomplishes solving the tension between bulk and boundary unitarity that other models in three dimensions suffer from. We study this theory at the chiral point, i.e. at the point of the parameter space where one of the central charges of the dual conformal field theory vanishes. We investigate the nonlinear regime of the theory, meaning that we study exact solutions to the MMG field equations that are not Einstein manifolds. We exhibit a large class of solutions of this type, which behave asymptotically in different manners. In particular, we find analytic solutions that represent two-parameter deformations of extremal Banados-Teitelboim-Zanelli black holes. These geometries behave asymptotically as solutions of the so-called log gravity, and, despite the weakened falling off close to the boundary, they have finite mass and finite angular momentum, which we compute. We also find time-dependent deformations of Banados-Teitelboim-Zanelli that obey Brown-Henneaux asymptotic boundary conditions. The existence of such solutions shows that the Birkhoff theorem does not hold in MMG at the chiral point. Other peculiar features of the theory at the chiral point, such as the degeneracy it exhibits in the decoupling limit, are discussed.