This paper is dedicated to the unique continuation properties of the solutions to nonlinear variational problems. Our analysis covers the case of nonlinear autonomous functionals depending on the gradient, as well as more general double phase and multiphase functionals with $(2,q)$-growth in the gradient. We show that all these cases fall in a class of nonlinear functionals for which we are able to prove weak and strong unique continuation via the almost-monotonicity of Almgren's frequency formula. As a consequence, we obtain estimates on the dimension of the set of points at which both the solution and its gradient vanish.
