As we proceed from categories to bicategories to tricategories to tetracategories, the coherence diagrams expand at an alarming rate, taking up a page, then 5 pages, then 51 pages. There is a shared view that an algebraic definition of $n$-category for high $n$ will be impractically enormous. While I agree with this opinion, I don't actually know any formal complexity result that specifies how quickly the coherence diagrams for an $n$-category blow up in scale with respect to $n$. One could perhaps look at associahedra for the associativity conditions, but this only handles one part of the problem. (I'm also not quite sure how one would define the 'complexity' of a diagram - perhaps by its number of objects and morphisms, for instance.)

Does anyone know any results or literature that give some insight into the precise expected scale of coherence diagrams in a weak $n$-category for general $n$?