In this work, we focus on the subject of nonlinear discrete self-trapping of S=2 (doubly-charged) vortices in two-dimensional photonic lattices, including theoretical analysis, numerical computation, and experimental demonstration. We revisit earlier findings about S=2 vortices with a discrete model and find that S=2 vortices extended over eight lattice sites can indeed be stable (or only weakly unstable) under certain conditions, not only for the cubic nonlinearity previously used, but also for a saturable nonlinearity more relevant to our experiment with a biased photorefractive nonlinear crystal. We then use the discrete analysis as a guide toward numerically identifying stable (and unstable) vortex solutions in a more realistic continuum model with a periodic potential. Finally, we present our experimental observation of such geometrically extended S=2 vortex solitons in optically induced lattices under both self-focusing and self-defocusing nonlinearities and show clearly that the S=2 vortex singularities are preserved during nonlinear propagation.