Is there a $3$-category in which natural transformations between functors of small categories are $3$-morphisms? I'm not sure if this question is well-posed, but what I'm getting at is that in, e.g. the category of groups, the $1$-morphisms are group homomorphisms. Then if we look at $\mathbf{Grp}$ from within $\mathbf{Cat}$, the $1$-morphisms are functors and the $2$-morphisms are natural transformations. But it feels like in something we ought to be three levels up. Does there exist a $3$-category in which this makes sense?
 A: The $2$-category of (small) categories $\mathbf{Cat}$ is a monoidal $2$-category under the cartesian product ($C \times D$), and so it could also be treated as a single-object (weak) $3$-category whose $1$-morphisms, $2$-morphisms, and $3$-morphisms are categories, functors, and natural transformations respectively.
Alternatively, we could instead use $C+D$ (the disjoint union), $C*D$ (the join, where there is exactly one arrow from each object of $C$ to each object of $D$), or $C \otimes D$ (the tensor product whose corresponding internal hom has arbitrary families of arrows $F(X) \to G(X)$ (without requiring naturality) as arrows $F \to G$) for the monoidal structure.
A: I am not sure what you are asking for. In the the  strict 2-category $\mathbf{Cat}$ of small categories whose objects are small categories, $1$-morphisms are functors and $2$-morphisms are natural transformations, we cannot make sense of non-trivial three morphisms.
We can remedy this by going into the next homotopy level $\mathbf{2Cat}$ whose

*

*objects are $2$-categories;

*$1$-morphisms are $2$-functors;

*$2$-morphisms are lax natural transformations;

*$3$-morphisms are modifications (morphisms between natural transformations).

If this is what you are asking for, I am glad to add more details to my answer.
