Let $\mathcal{C}$ be a $1$-category. One can show that the category of internal categories in $\mathcal{C}$, with internal functors and internal natural transformations produce a $2$-category.
I find this rather surprising, considering that an internal object in $\mathcal{C}$ is a functor (generally continuous and stuff) from a diagram category to $\mathcal{C}$, while internal morphisms are natural transformations between such functors. Thus, there naturally exists a structure of 1-category associated to internal objects, but a priori no 2-categorical structure. I tried to check if the concept of internal transformation could be encoded as a sort of "modification" using 1-cells only, but it seems to fail hard. This means that the 2-cells structure is "non canonical" and has to be put by hand. Hence my question: is there a maximal $n$ such that internal objects in a $1$-catetgory with their internal cells up to $n$ forms an $n$-category?
PS: I guess I would expect such $n$ to be 2 because the only "internal structure" of an object is given by its generalized elements (that is, arrows), but I feel confused now.
Technically there is no maximum. You could create ∞-categories from a 1-category, but anything higher than the initial 1-category would be trivial.