  We study the inverse boundary problem for a nonlinear magnetic Schr\"odinger
operator on a conformally transversally anisotropic Riemannian manifold of
dimension $n\ge 3$. Under suitable assumptions on the nonlinearity, we show
that the knowledge of the Dirichlet-to-Neumann map on the boundary of the
manifold determines the nonlinear magnetic and electric potentials uniquely. No
assumptions on the transversal manifold are made in this result, whereas the
corresponding inverse boundary problem for the linear magnetic Schr\"odinger
operator is still open in this generality.
