Yup, .308. The guns are readily available, the ammunition is available - Black Hills 175 gr match is a little less than a dollar a pop until you get smart and start handloading - barrel life is long, and recoil is moderate.
I shoot F/TR where only .223 and .308 are allowed at up to 1000 yards and a friend shoots Palma where only .308 is allowed at 800, 900, and 1000 yards. We have to work harder on wind estimation than the guys with the hotter guns, but we do ok and have a good time.