I have an Xbox controller that lets you easily pop off the buttons and sticks and replace them with new ones. The sticks actually just detach and you can push a new one on, so you don't need to open up the controller unless something internally goes bad. I think these are sometimes prone to "drift" and you might need to open them to re-center the sticks.

I didn't know until today that people also buy longer, differently shaped sticks for better aiming and control, which I don't exactly need since I mostly play platformers.