What is the most beautiful math result in your opinion?*
Personally, I usually like approaches that allow you to solve a problem in an indirect way.
My favorite result is probably the uniqueness of the Poisson equation (I don't see why this wouldn't hold for other differential equations too, but as a physics guy this is the area I saw it applied so I'm just going with that) as it allows for the method of images. I don't think there is any other result which when I first saw applied made me go WTF.
For similar reasons, I think the residue theorem is quite cool too. Who would have thought that to solve real integrals you would need to move off the real line and look for singularities elsewhere.
I mean, it's quite amazing if you think about how that must have developed historically.
An unrelated result which I quite like, which you introduced me to Chruser, are the Borwein integrals.