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Charged black holes in anti-de Sitter space become unstable to forming charged scalar hair at low temperatures T < Tc. This phenomenon is a holographic realization of superconductivity. We look inside the horizon of these holographic superconductors and find intricate dynamical behavior. The spacetime ends at a spacelike Kasner singularity, and there is no Cauchy horizon. Before reaching the singularity, there are several intermediate regimes which we study both analytically and numerically. These include strong Josephson oscillations in the condensate and possible 'Kasner inversions' in which after many e-folds of expansion, the Einstein-Rosen bridge contracts towards the singularity. Due to the Josephson oscillations, the number of Kasner inversions depends very sensitively on T, and diverges at a discrete set of temperatures {Tn} that accumulate at Tc. Near these Tn, the final Kasner exponent exhibits fractal-like behavior.