The Xiaomi Yi is under $100 and can do 1080p60.
The Mobius ActionCam is even cheaper than the Xiaomi, and it's true that can only do 1080p30, but the very low price and the extremely small size (matchbox) make it perfect to carry around all the time on a keychain, or install it on a small drone without much lift capability.
I have both these cameras. I made a helmet mount for the Xiaomi and I carry it sometimes when riding my motorcycle. The Mobius recently went up on a model rocket and captured a PoV of the whole flight.