Question: "What kinds of systems cannot be studied fully using Category Theory due to its inherent limitations?"
Answer: It is better to write down a list of fields "using" category theory "in an essential way" and then choose a field that is not on this list.
If you study objects such as real manifolds, complex manifolds or algebraic varieties you will in many cases study various types of cohomology groups, homology groups, Chow groups, algebraic K-theory and these groups are in many cases defined using the language of "derived functors" - this is a language originating in the 1950s and the work of many people. Hence if you study a field where cohomology and homology is important you will end up using "category theory".
Many of todays long standing conjectures in pure mathematics are related to problems with calculating these groups - conjectures such as the Hodge conjecture, the Standard conjecture (on Weil cohomology theories) and various conjectures in arithmetic (related to algebraic K-theory and special values of L-functions). The study of algebraic K-theory $K_n(X)$ and the Higher Chow groups include the Chern character:
The Chern character is an isomorphism between higher algebraic K-groups and higher Chow groups
$$ ch^m_i: \operatorname{K}_m(X)_{\mathbb{Q}}^{(i)} \cong \operatorname{CH}^i(X,m)_{\mathbb{Q}}.$$
The higher algebraic K-groups are defined in terms of the category of vector bundles on $X$, the higher Chow groups are defined in terms of the set of all closed integral subschemes of $X$.
Hence Chow theory is "more set theoretical", algebraic K-theory is "more category theoretical". Hence you can study these conjectures using "less category theoretical machinery" - then you study the Chow groups. Else you study algebraic K-theory.
Example: The Chern character above does not exist in "complete generality", hence if you want to use "less category theory" and use higher Chow groups to study "conjectures on special values of L-functions", you must first prove existence of a Chern character as above in greater generality. But in this process of generalization you will have to study algebraic K-theory and then it is inevitable: You must study some category theory in order to understand algebraic K-theory.
Why do we use divisors in algebraic geometry?
But: When you study algebraic geometry, much of the litterature is written in a "functorial language" (in the EGA, SGA book series and the stacks project), hence you will usually need some category theory when reading these books/notes.
Example: The Yoneda lemma is much used in algebra and algebraic geometry in defining objects such as the Hilbert scheme, Quot scheme and various other "moduli spaces". The Lemma is not a "deep lemma". It says that you may for any locally small category $C$ define the "functor of points" $h$
$$h: C \rightarrow Funct(C,Sets),$$
and that the functor $h$ "embeds" $C$ as a subcategory of $Funct(C,Sets)$. The Hilbert scheme is constructed using this Lemma. Does this mean you "use category theory" in an essential way in your work when you study the Hilbert scheme?
Example: If $X$ is a scheme and $E$ is a finite rank locally trivial sheaf on $X$, you may define the projective space bundle $\mathbb{P}(E^*)$ in two ways: You may define it using representable functors and the Yoneda lemma, or you may define it using a local trivialization of $E$ and glueing of schemes. In the case of the grassmannian functor you construct an open cover of representable sub functors and "glue functors". Hence in this situation you have a choice. You can choose a more direct approach and avoid using the functor of points and category theory. In many cases you have this type of choice: A direct approach and an "indirect approach" using the Yoneda lemma. In these cases where you do have such a choice: Does this mean you "use category theory" in your work? A "problem" with the Yoneda lemma in algebraic geometry is that you end up studying non small categories - categories where the objects form a class (and not a set). Hence you use non standard set theory. I believe there are quite many people that are using this construction without being aware of this fact.
Example: Whenever you study objects and have a notion of "isomorphism", this notion depends on which category you are considering. This does not neccessarily mean you "use category theory" in your work. What it means is that you must be careful which category you are working in, if you change the category your isoclasses change. As an example: Consider the case where you study complex quasi projective manifolds up to isomorphism. If you add the hypothesis that your manifolds are algebraic, then it follows you are studying smooth complex projective varieties. The same for coherent sheaves. If $E$ is a coherent sheaf on a complex quasi projective manifold $X$, and if $X$ happens to be projective it follows $E$ "is" projective (or: It is "equivalent" to a coherent algebraic sheaf $E^{alg}$ on $X$):
Motivating (iso)morphism of varieties
This does not necessarily mean you "use category theory" when you study coherent sheaves on complex quasi projective manifolds. There is for any complex quasi projective algebraic manifold $X$ an associated complex analytic manifold $X^{s}$ (see the above link) and for any coherent algebraic sheaf $E$ on $X$ there is a corresponding coherent analytic sheaf $E^s$ on $X^s$, and this correspondence is functorial. When you use this correspondence - does this mean you "use category theory" and that "category theory is essential" in your work? In this particular case it means you study smooth complex projective varieties, but where you include methods from complex analysis as a tool.
Example: If someone studies "Weil cohomology theories" and "algebraic cycles", this study involves much category theoretical machinery. Does this mean this person studies "category theory"?
https://en.wikipedia.org/wiki/Yoneda_lemma#Formal_statement
https://en.wikipedia.org/wiki/List_of_unsolved_problems_in_mathematics#Algebraic_geometry
