All examples so far involve only invertible k-morphisms for k≥3, i.e., they can be described as (∞-)bicategories. I would like to give an example of an interesting tricategory (with non-invertible 3-morphisms): The tricategory of conformal nets, as constructed by Bartels, Douglas, and Henriques.
This tricategory is used by Douglas and Henriques to give an algebraic description of string structures in the same vein as the existing algebraic descriptions of spin structures and orientations.
