Forgive me for the vagueness in the following paragraph, but I don't know how to communicate what I am thinking more formally.
If we have a definition for 1-categories (category) and a definition for 2-categories then surely we can abstract this to 3-categories, 4… all the way to infinity. From what I understand the main obstacle in defining the n-category is that two definitions of a category are equivalent if the category of said categories is equivalent, and so we are stuck in a loop. (I may not be communicating what I mean properly here..) But is it not the case that the infinite-category contains the n-category for all n, allowing us to stop the loop of cyclically defining categories? So my question is, what are the main obstacles in creating a definition for the n-category? Please note that I have had limit exposure to category theory (adjunction's, Yoneda…) and much less exposure to higher category theory, so a not overly technical explanation would be appreciated.
It's easy to define strict $n$-categories (meaning that composition is strictly associative at all levels and so forth): if you have a definition of strict $n-1$-category, you just define strict $n$-categories to be categories enriched over strict $n-1$-categories. The problem is first that many natural examples aren't strict, although when $n = 2$ you can hope to strictify them, and more importantly that when $n \ge 3$ many natural examples aren't strictifiable at all.
For example, whatever an $n$-category ought to be, every topological space ought to have a fundamental $n$-groupoid whose objects are points, whose morphisms are paths, whose $2$-morphisms are homotopies, and so forth. This is not naturally a strict $n$-groupoid (composition of paths is already not associative, but only associative "up to coherent higher homotopy"), and when $n \ge 3$ it generally isn't strictifiable: it turns out that strictifiability implies that Whitehead brackets vanish, and so already the fundamental $3$-groupoid of $S^2$ is not equivalent to a strict $3$-groupoid.
The difficulty of defining $n$-categories comes from figuring out all the coherence data and conditions necessary to define weak $n$-categories so that they capture, at the very least, $n$-groupoids as they arise in algebraic topology (see also the homotopy hypothesis). To get a sense of how hard this is to do "by hand," see, for example, Todd Trimble's definition of a weak $4$-category.
Even with strict $n$-groupoids, the naive way of defining what a functor is between these also fails to capture topological phenomena: for example, every group $G$ gives rise to a strict groupoid $BG$ with one object and automorphisms $G$, and every abelian group $A$ gives rise to a strict $2$-groupoid $B^2 A$ with one object, one morphism, and $2$-automorphisms $A$. The "correct" set of equivalence classes of morphisms $BG \to B^2 A$ is the second cohomology group $H^2(BG, A)$, but the naive description of a functor between strict $2$-groupoids won't get you this. The problem is that functors also cannot be taken to strictly preserve composition, etc. but must also come with coherence data and conditions.