When i hear "Big Game" I think Cape Buffalo, or other large African creatures, but you seem to be implying North American game... just get whatever caliber you want in whatever fits you. 300WM or 300 RUM will drop anything in North America just fine. Whether you spend $400 or $4000 on a factory rifle for hunting big game, the results will be similar.
If it's pretty furniture you want, buy a Weatherby. They have some big ass cartridges such as their , 340, 378, 416 and 460's. They are nice to look at and hold, but they're no more accurate than a $500 Savage or Remington in a similar Caliber. CZ, for less money than a weatherby also has some pretty big stuff available.
If it is ultra accuracy you want, buy a Remington 700 in anything up to 338 and re-barrel it with a custom barrel. It will cost less and be way more accurate. 338RUM will kill almost anything.