For the purposes of this post we will use the one hom class definition of a category.
Note that a functor $F:\mathcal{C}\to\mathcal{D}$ between categories is a pair of functions $F_0:{\bf Ob}_\mathcal{C}\to{\bf Ob}_\mathcal{D}$, $F_1:{\bf Hom}_\mathcal{C}\to{\bf Hom}_\mathcal{D}$ satisfying coherence conditions, and a natural transformation $\alpha:F\Rightarrow G$ between parallel functors $F,G:\mathcal{C}\rightrightarrows\mathcal{D}$ is a function $\alpha:{\bf Ob}_\mathcal{C}\to{\bf Hom}_\mathcal{D}$ satisfying coherence conditions, where we usually write $\alpha(X)=\alpha_X$.
Taking things one dimension up and using the (admittedly cumbersome) nuts-and-bolts definition of a bicategory $\mathcal{C}$ as consisting of one object class, one class of $1$-cells and one class of $2$-cells together with the obvious functions between them to make them into a bicategory (we can formalize weak associativity using a function from an appropriate pullback of $1$-cell classes into the $2$-cell class satisfying appropriate coherence conditions).
Taking this view, we have that:
A pseudofunctor $F:\mathcal{C}\to\mathcal{D}$ is a triplet of functions $$F_0:{\bf Ob}_\mathcal{C}\to{\bf Ob}_\mathcal{D},$$ $$F_1:{\bf 1-Cell}_\mathcal{C}\to{\bf 1-Cell}_\mathcal{D},$$ $$F_2:{\bf 2-Cell}_\mathcal{C}\to{\bf 2-Cell}_\mathcal{D},$$ satisfying appropriate coherence conditions.
A pseudonatural transformation $\alpha:F\Rightarrow G$ between parallel pseudofunctors $F,G:\mathcal{C}\rightrightarrows\mathcal{D}$ is a pair of functions $$\alpha_0:{\bf Ob}_\mathcal{C}\to{\bf 1-Cell}_\mathcal{D},$$ $$\alpha_1:{\bf 1-Cell}_\mathcal{C}\to{\bf 2-Cell}_\mathcal{D},$$ satisfying appropriate coherence conditions, where we write $\alpha(X)=\alpha_X$ and $\alpha(f)=\alpha_f$.
A modification $\mathcal{M}:\alpha\nrightarrow\beta$ is a function $$\mathcal{M}:{\bf Ob}_\mathcal{C}\to{\bf 2-Cell}_\mathcal{D}$$ satisfying appropriate coherence conditions, where we write $\mathcal{M}(X)=\mathcal{M}_X$.
For an arbitrary natural number $n$, we could define an $n$-category $\mathcal{C}$ as consisting of $n+1$ classes $\{\mathcal{C}_i\}_{i<n+1}$ of cells (where we are now referring to ${\bf Ob}_\mathcal{C}$ as $\mathcal{C}_0$), together with the 'obvious' functions between them to make them into a weak $n$-category. We could then define an $n$-functor $F:\mathcal{C}\to\mathcal{D}$ to consist of $n+1$ functions $\{F_i:\mathcal{C}_i\to\mathcal{D}_i\}_{i<n+1}$ satisfying 'appropriate coherence conditions', an $n$-natural transformation $\alpha:F\Rightarrow G$ to consist of $n$ functions $\{\alpha_i:\mathcal{C}_i\to\mathcal{D}_{i+1}\}_{i<n}$ satisfying 'appropriate coherence conditions', an $n$-modification $\mathcal{M}:\alpha\nrightarrow\beta$ to consist of $n-1$ functions $\{\mathcal{M}_i:\mathcal{C}_i\to\mathcal{D}_{i+2}\}_{i<n-1}$ satisfying 'appropriate coherence conditions', ... , and a 'natural $n+1$-mapping' to be a function $m:\mathcal{C}_0\to\mathcal{D}_n$ satisfying appropriate coherence conditions ($n$ categories naturally have $n+1$ total notions of morphism between them and their morphisms taking this view (modulo differences in weakness/strictness), as is the case in low dimensions).
Is there any existing literature on higher $n$-categories defining them this way, or in a similar fashion?
The motivation is that, taking a recursive view, we could try an inductive attempt at proving a coherence theorem (in the sense of Gordon/Power/Street and Gurski) for general weak $n$-categories into '$n$-Gray Categories'. Specifically, we could assume that for some $n$ the coherence theorem holds, then prove that this implies it in dimension $n+1$, and since we know it holds in low dimensions this would settle the case for all $n$.
Obviously, phrases like 'obvious' and 'appropriate coherence conditions' sweep an increasingly mountainous amount of definitions under the rug each time we move up a dimension, so this imprecise phrasing of the definitions involved may be fraught with peril -- if there are any glaring issues I have missed, I apologize in advance.
In order for the recursive bit to be useful, we have to use the notions of 'category', 'functor', etc. defined at the previous $n+1$ stages in our definition of 'category' at the $n+1$th stage, which would naturally lead us towards the 'high tech' (or 'typed') recursive definition of an $n+1$ category as consisting of an $n$ category for each pair of objects, an $n$ functor for each triplet of objects, ... , and a 'blah mapping' for each collection of $n+2$ objects (think associator natural isomorphism for each quadruplet of objects in a bicategory).
Taking this view, we could define a 'weak $n+1$-category' as having appropriately 'weak' component $n$-categories/functors etc., then assume coherence theorems hold in dimension $n$ that let us replace these weak components with appropriately strict ones and proceed to attempt to show that the resulting $n+1$-category/functor/etc. is also appropriately strict.
The issue with using this definition is that we lose the clean pattern outlined above for what constitutes the various notions of 'mapping' between $n$-categories and their morphisms. There is another pattern for this case, but I don't typically use the typed definition enough to write it down without thinking about it for some time first (perhaps some kind hearted type-theoretical category theorist here will assist me).
In the definition outlined above I think we could still define an $n+1$ category in the nuts-and-bolts way and then show that we have component $n$-categories for each pair of objects, then assume and use coherence as we would in the typed case.
In any event, the main gist of this post is the one in the title and the highlighted portion -- has anyone ever defined higher categories in a manner similar to this, perhaps with an eye towards inductively proving coherence for finite $n$-categories?
