Robin Green is right, I think, in saying that "this should ... be tagged with foundations".
Tim Porter hit on an essential part of the equation for new discovery in the sciences, maths included: "ripeness" or "readiness". As a young child passes through the various necessary developmental stages, so it builds a repertoire of active concepts that both inform and manage its interactions with the external world. At some stage it begins producing language, not just hearing it. Once it's done that, the child can proceed to holding conversations, to understanding stories that involve dialogue, to reading, and to writing. But each stage is necessary to the subsequent ones; not even a "genius" (*) can completely skip all the intervening stages to pass from hearing language directly to writing it.
(*) A digression on what we mean by "genius": In fact, what we consider "genius" is often nothing more than a degree of insight which enables its possessor to move so swiftly through intervening stages that it may appear as though those stages were skipped. Look, for example, at Évariste Galois, and his amazing insights into group theory: he was clearly (and fortunately for us) "before his time"; yet he could not have produced his results without first passing - rapidly! - through the necessary intervening stages that included the current state of play in group theory; nor without insightfully applying those ideas to new ground.
Similarly, as we study the mathematical aspects of the external world, we can only develop concepts when a suitable foundation is in place. In this sense, abstract algebra is foundational to category theory, differential equations to analysis, and arithmetic to algebra - not the other ("logical") way round. So when we're looking for foundational ideas in maths, let's remember that a suitable foundation for deducing other ideas is usually only arrived at very late in our process of understanding those other ideas, by a considerable amount of abstraction and much sweat spent on proving that that foundation (or axiom schema, or theory) suffices and is necessary (only logically!) for those deductions.
In short, I'm saying to user1613 that discovering something more "fundamental" to maths than category theory (CT) will probably only happen after we've had plenty more experience with using CT and applying it in places not covered by the ideas that generated it (i.e. outside algebraic topology). Or, if we're lucky, it may happen sooner than otherwise "normally" to be expected if some genius happens along with amazing (but not impossible) insights that connect CT to hitherto unsuspected areas of study. Care to combine CT and molecular biology, anyone? ;-)
And of course, the genius that does happen along will be an "outsider", in other words, somebody who's not too close to the problem to begin with. Which will put many dedicated researchers' individual and collective noses out of joint. Remember David Hilbert's famous Hilbert Programme? Everybody believed that here were a century's worth of serious problems needing elucidation. But it only lasted about 30 years, when along came a Young Kurt (Gödel to be precise), with some devastating news about undecidable problems in proof systems such as Principia Mathematica, Bertrand Russell's crowning achievement of research into mathematical foundations. Ironic, that last ...
One more point, and I'm done. And that is that fundamental advances are usually disruptive, not the result of an assembly-line, production-oriented approach to making progress; by definition, such a "steady as she goes" mindset can only ever produce more of the same. User1613, you're looking for a big idea in the grand, unifying tradition of CT; but CT itself sprang from the difficulties of making sense doing algebraic topology, not from the impetus to produce some sort of Unified Field Theory for maths. I'm willing to bet that if you, for example, dedicate yourself to finding "the thing more abstract than CT", you'll only make any progress by first making several totally unexpected discoveries about apparently unrelated things.