"Big game" is specified so I'm omitting the obvious rimfire(s). The first two below are in the safe after much thought.

243 Montana (twisted by PN for 105 hornys exclusively) under a 3-9x42 that tracks.

30-06 Hawkeye All-Weather under a Leupy 4x. (1:10 handles 220's).

Don't know on the third. Hard not to want a rattle canned 458 Win or 375 H&H Mauser with a receiver sight after reading Shoemaker's stuff. In reality, the '06 will likely handle what I need of the big stuff and I'd be better off going the other direction with a handy 223 bolt twisted for 75's.