To break the joystick actually requires a considerable amount of force/rage. I broke mine several months ago by slamming the controller face first into the floor, and then proceeding to stomp it a few times. Why did I do this? Well, for anyone who has attempted to get the "Mile High Club" achievement in Call of Duty 4 probably knows what I'm talking about. After about 25 times of almost completing it, frustration may ensue. The ironic part is, after I bought a new controller, I beat in 2 tries. Oh, well.
The controller still functions properly, with exception to the obliterated right joystick. The thumbstick was completely undamaged which is a good thing because that meant I didn't have to buy a new one.
If you happen to be one of those people who have a broken thumbstick, all you need to do is buy, or procure one from another, more severely broken controller and follow this Instructable up to STEP 2.
You can buy replacement thumbsticks here:
You can choose between gray or black thumbsticks and even PS2/3 style sticks.
You can buy the replacement analog joysticks from SparkFun.com here:
They only cost $3.95 so it's a lot better than $50 for a new controller.
For those with broken analogs, follow all the steps.