As I see it, it there is no lower bound on the number without placing restrictions on the "expressions" involved. Specifically, it would seem that any such expression would have to be a map from the set of possible universes to the positive whole numbers. Take any such map, call it θ. Then let φ = θ + 1 so that φ(U) = θ(U) + 1 for some universe U. Then φ(U) > θ(U). 

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