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Self-assembly materials are traditionally designed so that molecular or mesoscale components form a single kind of large structure. Here, we propose a scheme to create “multifarious assembly mixtures,” which self-assemble many different large structures from a set of shared components. We show that the number of multifarious structures stored in the solution of components increases rapidly with the number of different types of components. However, each stored structure can be retrieved by tuning only a few parameters, the number of which is only weakly dependent on the size of the assembled structure. Implications for artificial and biological self-assembly are discussed.