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We introduce a hierarchical classification of theories that describe systems with fundamentally limited information content. This property is introduced in an operational way and gives rise to the existence of mutually complementary measurements, i.e. a complete knowledge of future outcome in one measurement is at the expense of complete uncertainty in the others. This is a characteristic feature of the theories and they can be ordered according to the number of mutually complementary measurements, which is also shown to define their computational abilities. In the theories multipartite states may contain entanglement, and tomography with local measurements is possible. The classification includes both classical and quantum theory and also generalized probabilistic theories with a higher number of degrees of freedom, for which operational meaning is given. We also discuss thought experiments discriminating standard quantum theory from the generalizations. [ABSTRACT FROM AUTHOR]