The post says three calibers (cartridges) so I' ll assume rifle cartridges for hunting big game.

North America

.260 Rem
.270 Wby Mag
.338-06

The World

.270 Win
8x68S
.375 H&H

Both

6.5x55 Swedish
.300 Wby Mag
.375 H&H

But if held to it, I' d say two rifle/cartridge combos are all you need to hunt big game anywhere in the world:

6.5x55/.270 Win/7x57 (take your pick for all hooved game)
.375 H&H (for all dangerous game)