Photonic lattices with saturable nonlinearity exhibit an intriguing dual behavior: they recover a second linear regime at high light intensities, offering a unique platform to study nonlinear topological effects beyond perturbative regimes. We numerically investigate the nonlinear continuation of topological edge states in the paradigmatic Su-Schriefferâ€"Heeger (SSH) chain with saturable nonlinearity. By computing exact stationary nonlinear edge modes and following their evolution across bifurcations and stability transitions, we reveal distinct nonlinear regimes that govern the dynamics of edge-localized excitations. In each regime, we evaluate two canonical signatures of bulk topologyâ€"the mean field displacement and the Zak phase (via projector methods)â€"extending these concepts to the nonlinear setting. Our results uncover how nonlinearity deforms, preserves, or disrupts the topological character of the system, and we identify parameter regions where a nonlinear bulkâ€"edge correspondence remains approximately valid. This study offers a comprehensive picture of nonlinear topological phases in saturable photonic lattices, bridging low- and high-intensity limits and highlighting the emergence of nonlinear-protected edge modes with topological origin.