Well, since the OP wrote "big game," I'll dispense with smaller calibers, save one. I've never fired it, but I'm becoming increasingly interested in the 6.5mm Creedmore, mostly because of its outstanding long range accuracy potential. Should be really great for antelope at 1,000 yards and such.
For a middling cartridge, I prefer .308. I have a beautiful FN Mauser Sporter Deluxe in that caliber and it is a dead on accurate sub-moa bolt gun. .308 is sufficient really for most any north American big game. But often people want a bit more punch. And for that, I have a particular preference:
For the really big game -- moose and grizzly and the like, I don't really see the need for a very large bore cartridge. I prefer a smaller caliber where the bullet is traveling at a very high velocity. So, I would go with the .338 Lapua. A 250 grain soft-tipped bullet, traveling at over 3,000 feet per second, with a muzzle energy in excess of 4,800 lbf -- that should stop just about anything I think I'd ever want to hunt.
Now, if some targets were something less than big game, after some deliberation, I'd change one gun and trade out another. For the smallest caliber, I'd select 22-250, mostly because of the outstanding experiences I've had with this round. I don't want to discard the Creedmore, so I'm inclined to replace my .308 with it, but I do this with some trepidation because I really like shooting the .308. I'd also keep the .338 Lapua for a high-powered round.