In no particular order:
.22 LR--cheap, economical, and fun with enough accuracy and potency to fill the stew pot. No collection is complete without at least one. Mine is a 10/22.
7.62x39--my favorite of the "intermediate" powered assault rifle rounds. Powerful enough for deer sized game, the round comes into its own in semi-auto carbines like the AK or SKS. Gets the job done out to 300+ yards and doesn't abuse the shooter doing so.
7.62x51/.308 Win--Powerful, efficient, versatile, and effective. Combining legendary accuracy with most of the utility of the .30-06 in a shorter platform, the cartridge can do anything a rifle might be needed for from defense to target and competition to large game. I love my M1A.
.270 Win--My first hunting rifle. Deer cartridge extraordinaire and also an excellent choice for North American big game from pronghorn to moose.
7mm Rem Mag--The trajectory of the .270, the recoil of the .30-06, and downrange energy and momentum approaching that of the .300 mags-what's not to love? Highly efficient projectiles and a recoil level most experienced shooters can adapt to have made this among the most popular magnums ever for good reason. Whether for deer, elk, or the 1000 yard range, the 7mm Rem Mag is a highly useful cartridge with very few faults.
Honorable mention to the .50 BMG and the .30-06. The .50 is more fun than useful for most of us, but fun it is and that is reason enough. And the "ought-six" is an American icon probably still unmatched in overall versatility and utility.