I'm not an expert in this area, but I've heard that algebraic topologists run into congruences quite often.  For example, the [stable homotopy groups of spheres][1] are almost always finite abelian groups, and are often studied by examining the Sylow subgroups with $p$-specific toolsets.

On the subject of infinite series, one encounters congruences working with the spectrum $tmf$ of [topological modular forms][2].  It admits a map from $\pi_0$ to the ring of level 1 modular forms with integer coefficients that is an isomorphism after 6 is inverted.  The image satisfies some congruences mod powers of 2 and 3 that are akin to the Borcherds congruences satisfied by theta functions.  For example, the cusp form $\Delta$ does not appear in the image, but $24\Delta$ and $\Delta^{24}$ do appear.


  [1]: https://en.wikipedia.org/wiki/Homotopy_groups_of_spheres
  [2]: https://en.wikipedia.org/wiki/Topological_modular_forms