I think that your intended question is too vague and that your attempts to make it precise probably don't capture your intent (assuming I understand your intent correctly).

Suppose I do a gigantic (but finite) computer calculation to produce an approximate numerical solution to a PDE, or to solve the Boolean Pythagorean triples problem. That is certainly "concrete" mathematics in some sense of the word "concrete." Technically, the validity of the computation can be expressed in set theory or in any halfway-plausible candidate for a foundation of mathematics. I suspect, however, that even if it were possible to encode the gigantic finite computation as some incomprehensible formal string in the "language of category theory" in some technical sense, you wouldn't consider such a monstrosity to be a satisfactory category-theoretic "expression" of that piece of "concrete mathematics."

If I'm wrong, and you'd be satisfied with that, then it seems your question is whether it's possible to develop categorical foundations for mathematics without piggy-backing on standard foundations. That's a more standard question which has been much discussed.

On the other hand, if you're asking how much mathematics can be "naturally" expressed in category-theoretic language, then we're back to a vague question. There are certainly areas of mathematics where there's no obvious concept of a "morphism." Consider a proof that merge sort correctly sorts a list of items using $O(n\log n)$ comparisons. If you want to express this in category-theoretic terms, what's your category? What are the morphisms? Any attempt to shoehorn this basic result in the theory of algorithms into category-theoretic terms is going to seem artificial. This is perhaps an extreme example, but even for a less extreme example, I have to wonder why one would want to express something in category-theoretic terms if it doesn't seem natural to do so. And if you are not interested in any shoehorning, and just want to know how much stuff fits naturally into category theory, then we're back to Andrej Bauer's 67.8% comment.