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).

So it would be dependent on restrictions placed on the set of allowable mappings. One obvious way to do this would be to index the set of possible states that the universe could be in. If there are a finite amount of states, then the amount of states is the largest such number. If there are denumerably infinite states, then there would be no largest integer. If there are more states than that, this technique doesn't make sense.