Okay then, here's another. 2-Hilbert spaces as a categorification of Hilbert spaces, and the categorified Gram-Schmidt process (which I first learned from James Dolan).

This may be used to derive a $\mathbb{Z}$-linear basis for the representation ring of $S_n$ that consists of permutation representations, hence a combinatorial alternative to the basis consisting of irreducible representations. The reference above sketches how this works in the case $Rep(S_4)$.