In Higher Topos Theory, Lurie argues that the coherence diagrams for fully weak $n$-categories with $n>2$ are 'so complicated as to be essentially unusable', before mentioning that several dramatic simplifications occur if we step up to $\infty$-categories but require all higher morphisms to be invertible.
What are some examples in nature (topological quantum field theory, string theory etc.) of notions that are best understood as a weak $(\infty,n)$-category, or simply a weak $n$-category, for 'large' $n$? (something which is naturally a 'fully weak' $\infty$-category would be interesting too)
Bordisms between $n$-dimensional manifolds have the former structure naturally and serve as the canonical example coming from TQFTs; are there any others?
(I believe the present question differs from this one, although the answers may overlap, as I'm looking for examples of $\infty$-categories that are 'motivated by nature' in the above sense and already believe higher categories are useful.)
I am also interested in natural occurrences of $\infty$-categories or weak $n$-categories for 'large' $n$ in areas of mathematics outside of category theory, so please share any good ones that occur to you. The use of $2$-category theory for the Galois theorem of Borceux and Janelidze wouldn't quite qualify, but if there is any area outside of category theory that seriously uses weak $3$-categories I would be interested to hear about it (invariants of $3$-manifolds?). This is why 'large' is in quotations; examples with $n>>1$ would be great, but I consider $3$ 'large' in certain contexts where $2$ isn't.