We investigate a large class of infinitesimal, but fully nonlinear in the field, transformations of the Galileon and search for extended symmetries. The transformations involve powers of the coordinates x and the field π up to any finite order N . Up to quadratic order the structure of these symmetry transformations is the unique generalization of both the infinitesimal version of the standard Galileon shift symmetry as well as a recently discovered infinitesimal extension of this symmetry. The only higher-order extensions of this symmetry we recover are ("Galileon dual" versions of) symmetries of the standard kinetic term.