Something like a GoPro, http://www.walmart.com/cp/GoPro-Camera/1095238
might be what you're looking for? (they can be controlled via phone apps too)
I think there's a new `Hero 4`due out soon, so the present models should reduce in price.
If it's purely a stills camera you want, you will presumably need it to be weather/waterproof so maybe look at something like the Nikon COOLPIX AW100.
btw,He who dies with the most toys, WINS!Nikon D800E & D700 bodies + Nikon 200-400mm F4 VR1, 50mm F1.4G, 16-35mm f/4G VR, 105 F2.8 VR macro, 70-300mm lenses. A couple of filters, Giotto tripod & ballhead. Lowepro Slingshot 302 AW