4 centerfires eh?
The man said that a .22 rimfire and shotgun are a "given", so that rules out listing them, as the question assumes them. Rifles only, centerfire only. Also, the question is North America, not Africa, so that leaves out my CZ 550 American Safari Magnum in .458 Lott choice. But it does include Alaska/Canada, so that's gonna necessitate one big game thumper. And self-defense was left out - the question is for hunting both game and varmints. That leaves me with:
1. Varmints and small game: something in .223 rem - either my AR15 in .223 or my NEF Handi-Rifle in .223 - probably take the Handi-rifle for max accuracy, assuming I had the self-defense base covered (as the question assumes). If I had the budget, it would be a CZ 527 varmint laminated, or a Remington XR100 instead of the Handi.
2. All-purpose big-game hunter: something in 6.5x55mm (or possibly its ballistic twin, the .260 rem) - either a Win 70, CZ 550, Tikka T3, or Howa 1500. Light version... sporter weight short barrel (20-22"), light-ish stock, light scope & rings - a 'mountain rifle'.
3. Second/backup all-purpose big-game hunter: easy, my Armalite AR10 in .308 win. Up to 180 grains with fast follow up. Double-tap your elk!
4. Very large game thumper for canada/alaska: this one is tougher - either .338 winmag, .35 whelen, or .375 H&H mag. Probably take a Remington 750 semi-auto in 35 whelen. If it's coming to hurt me, I want a big semi-auto.