Okay, we seem to all agree that everyone should have a 12 gauge and a 22LR, so here's my take from there:
1) A 223, preferably in a "modern sporting rifle." These days I recommend you consider dual purpose functionality needs, and I won't say anything more than that. Pretty sure I'm in the right room to say everyone here knows what I mean.
2) A 308 - again consider the MSR or something like it (same reason as above) but a bolt rifle is also good - never know what the future might bring, and knowing what ammo will be available might become a very critical aspect. The 308 is no longer the king of long range accuracy, but a couple of generations of military snipers can vouch for it's capabilities. Never argue with a proven track record! Remember that string about mil-spec brass?
3) A 30-06 bolt gun - back in the day the mantra was that the '06 could be used for any hunting in N. America, and all I can say is that I've seen it take everything from coyotes to bison (Okay, did not actually witness moose or bison but know the guy who did it!) So why include this if you already have the 308 on the list? Several simple answers - brass everywhere, and I have to admit that with any respectable big game pill accuracy is easy. Plus it really does handle deer, elk and moose about as well as most shooters can put it on target as long as you pick the right bullet. Easy to load it up or down for whatever purpose, and modern powders & pills continue to make it better as time goes by...115 years and still gittin' it dun? If I honestly had to pick just one this would be pretty hard to walk away from.
Okay, those were the easy 3...now my 2-cents-woth on some more specialized needs:
4A) Call this a tie: For true long range work: I love the 6.5 Creedmoor because you shoot all day without breaking the bank or your shoulder. Only caveat is that I'm hesitant to tell anyone it's an all-around hunter. PH - deer- sheep and such - no worries. Handles nicely even in a flyweight rig. But if you start talking about bigger and tougher critters whether it's elk or a 350-lb black bear...well, I think you are better off with more horsepower. In a perfect world it's all good, but I personally am not spending a couple grand on my own self-guided elk hunt to count on getting a perfect shot.
Which leads us to the tie:
4B) Again for true LR - 300 RUM. Bench rest accuracy with Mack truck horsepower. If you really want to shoot elk at 600+ yards, this is a great choice. Launch a 200-210 grainer at 32-3300 FPS and the results way out there are something to marvel at. Drawbacks - no free lunch here - the recoil is not to be taken lightly. Neither is carrying around a ten-lb. rig when you are walking in for miles, the altitudes are up there and flat ground is scarce. Go lighter and you might risk a detached retina. I am told that is no fun!
5) I admit I am really not qualified to critique the big bores as I've never used anything bigger than a 375 H&H on game. I've never hunted Africa so I would defer to those who have...that said, I think if you really want to use a big bore with some longer range capabilities the 375 RUM has received some decent reviews.
Honorable mentions: 300 Win Mag & 7mm Rem Mag
Hard not to give some love to the 300 Winmag and the 7mm Rem Mag. I definitely enjoy both! Performance-wise, 300 WM is like a 30-06 on steroids. Exceptions - it doesn't load down for lower recoil very well, and working with bullets lighter than a 150 can be a challenge...can you say temperature/pressure sensitive? But if you use the right 150-168 then even coyote hunting is still on the table. Some would say that's overkill, but anyone who has tried taking coy dogs at 400 in serious wind is nodding yes! Main reason I chose the 300 RUM over this cartridge is the performance with the heavy pills. But there are two significant advantages: From experience the recoil of 300 WM is the practical limit for many shooters, and this cartridge has been available in every style of hunting rifle action you can name. Even lever and pump-action fans can play the game.
As for the 7mm Rem Mag - if brass & ammo were as common as 30-06 this would have tied for that #3 slot. Not much it cannot do - Fairly easy to load up or down with accuracy for most hunting apps, much more so than the 300 Win Mag. Select the bullet to match your target and hunting conditions, and results are excellent. If the deciding factor between 30-06 and 7mm Rem Mag comes down to long range, many would pick the 7 and it would be VERY hard to say they are wrong! Not quite as much raw power as the 300 WM but close enough that most would call it a wash. Only real drawback I can think of is that when handloading, many powder/primer/bullet combos are pretty touchy - seat the pill .005 farther in or out, temp goes up or down, and you have flyers.