The Littlewood-Richardson coefficients. (Or, if one wants a single object, as per the rules of the game: the representation ring of $S_n$ or $GL_n({\bf C})$. Or the ring of symmetric functions. Or the cohomology ring of the Grassmannian with the Schubert variety basis. Etc., etc.)
On the one hand, the Littlewood-Richardson coefficients have fairly simple geometric descriptions (using such combinatorial gadgets as Young tableaux, honeycombs, or puzzles), but on the other hand obey a number of deep recursive properties. (See for instance my Notices article with Allen Knutson on one aspect of these coefficients.) Last, but not least, they are connected to an amazing number of areas of mathematics (see e.g. Fulton's survey article).
