My personal favorite is Multiple Zeta Values $$ \zeta(s_1,\ldots,s_d) = \sum_{n_1>\ldots>n_d} \frac{1}{n_1^{s_1}\ldots n_d^{s_d}} $$ They appears in relation with
- Quantum groups (they are coefficient of Drinfeld's KZ associator)
- Deformation quantization (Kontsevich's formula for the affine space)
- Feynmann diagrams (a large class of diagrams evaluate to MZV's)
- Kashiwara-Vergne conjecture (representation theory of Lie groups)
- Modular forms (Zagier noticed that the space of relations in depth 2 is canonically isomorphic to the space of cusp forms on $SL_2$ through their period polynomials)
- Moduli spaces of curves of genus 0 $\mathcal{M}_{0,n}$
the list goes on and on... the reason for all this lies in the theory of mixed Tate motives over $\mathbb{Z}$.
