My Vote - The .300 Win. Mag.
I would pick a .308 caliber cartridge for the simple reason that there is a vast variety of bullets available for this caliber.
The .300 Win. Mag. can do everything that the .308 Win. and 30-06 Spfld. can do, and then some. With the right bullets you can extend the .308's effective range by another 200-300 yds. If you hand load, you can achieve muzzle velocities that range from the 30-30 on up to nearly the .300 UltraMag. With the right barrel twist, you can load 240 gr. VLD bullets and nearly equal the .338 Lapua Mag. performance.
The .300 Win. Mag. is plenty accurate, winning plenty of 1000 yd. matches in years past, and is still a popular cartridge for "F-Class" competition. There are cartridges that have eclipsed it, but they are more specialized than the .300.
Many of you have espoused the .22 LR as the cartridge of choice. While I have slaughtered cows with a .22, the brain shot was taken at 3 feet, WAY closer than I would like to let a bear or cape buffalo get. At longer distances, and without the situation control you have with a domesticated animal, the .22 LR is seriously underpowered for big game animals.
Understandably, the .300 Win. Mag. is overpowered for small game. Even loaded down to the minimum, hitting a rabbit anywhere but in the head will leave you with only little mangled chunks for the stew pot. But if you have to, shoot them in the head! Better restricting your small game shots to head shots than your big game shots, right?
Finally, it is a very popular big game hunting cartridge. If you happen to get separated from your ammunition while traveling to your hunting destination, you can usually find factory ammo at a store. Hey, there is probably even someone else in camp shooting a .300 Win. Mag. who would loan you a handful.