If we're talking about hunting elk, the minimum I've carried was a Crosman 760 .177 cal pump. My step-brother was carrying his Daisy Red Ryder though. I'd say the Red Ryder is a good minimum for elk hunting, but the plastic-stocked Daisy is lighter and shorter, thus better for field carry.
The elk killing was on my step-dad. He was carrying a 7 RM with factory Remington ammo. Seemed to work okay.
