It turns the cap on and off!

It is mereley a decoration, a way to hide where several pieces are joined. Not all caps have a button.

A form of torture!
So when people slap u on the top of your head it hurts!

spare button in case the fly in your pants loses a button

It hides the point where all of the panels of the hat meet. Some hats just connect there and do not have a button, but in my opinion the button looks much neater.

It's for break dancing when the guy spins on his head.

Probably to hide the spot where all the seam stitching meets up. It just finishes it off & makes it a little tidier without having to go to the trouble of trying to hide the stitches in the lining.

to hold the material together.

I don't think it has a purpose besides holding the hat together. It would look bad if you saw the hat being held together i think it just covers it up.

Push it and find out.

going to show my age but there was a show on HBO back in the days where they had sniglets. They were words you could not find in the dictionary. They called it a squatcho

To conceal the gathering of material.