For the one rifle and ALL NA game, for practicle sense, I picked the .375 H+H. It is more than adequate for any NA game when you include the big bears and BIG moose like in Canada. The trajectory is impressive compared to most any other selection. Bullet choice in good in factory loads and the most important is the ability to find the ammo most anywhere you can buy ammo. Otherwise, I would be in the range of the 9.3x62. Usually lighter rifles, but with plenty of power and penetration for any NA game with the right bullet. If you exclude Big brown bears and Polar bear (though many have been taken with less powerful cartridges), then I would be in agreement with Daryl and 9.3 with the 9.3x62, 9.3x57 and the I would include the 8x57 in proper loadings. The issues are of finding and buying a particular caliber cartridge when I can't bring my own is a concern for me.
Having said that, I would love the challenge of using my original .45-70 Rem. Rolling Block to do the same. i just received a new batch of .460 dia 500 grain RN cast bullets that I will load and test in the .45-70 and the .577/.450.