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To understand why molecular evolution turned out as it did, we must characterize not only the path that evolution followed across the space of possible molecular sequences but also the many alternative trajectories that could have been taken but were not. A large-scale comparison of real and possible histories would establish whether the outcome of evolution represents an optimal state driven by natural selection or the contingent product of historical chance events1; it would also reveal how the underlying distribution of functions across sequence space shaped historical evolution2,3. Here we combine ancestral protein reconstruction4with deep mutational scanning5,6,7,8,9,10to characterize alternative histories in the sequence space around an ancient transcription factor, which evolved a novel biological function through well-characterized mechanisms11,12. We find hundreds of alternative protein sequences that use diverse biochemical mechanisms to perform the derived function at least as well as the historical outcome. These alternatives all require prior permissive substitutions that do not enhance the derived function, but not all require the same permissive changes that occurred during history. We find that if evolution had begun from a different starting point within the network of sequences encoding the ancestral function, outcomes with different genetic and biochemical forms would probably have resulted; this contingency arises from the distribution of functional variants in sequence space and epistasis between residues. Our results illuminate the topology of the vast space of possibilities from which history sampled one path, highlighting how the outcome of evolution depends on a serial chain of compounding chance events.