The "unreasonable effectiveness of math" is most evident in physics - dualities, mirror symmetries - and my personal favorite, Einstein's being able to pull Reimannian geometry "off the shelf" to describe relativity. For me, it is breath-taking.
My favorite is Maxwell's equations ... he added in the "displacement current" (which correlates changes in the electric and magnetic fields), allegedly for reasons of symmetry, not because he'd actually seen something that needed describing.