Here's my all around North America list since Africa isn't in my forseeable future, but would IMHO, work well in Africa.
I also believe in a lot of cases, improvements in ammo and bullet technology/quality make addition of some calibers unnecessary. I'll explain in my list.
.223 Rem. A no-brainer, official NATO caliber, accurate. Ammo plentiful and cheap in a variety of varmint, plinking and anti-personnel loads. If you need more range, sneak up a little closer.
.30-06 Sprgfld. Probably more commercial loads available for this caliber than any other. Stoked with standard pressure/velocity ammo for general purposes. Federal High Energy and Hornady Light Magnums moves the '06 into .300Mag territory for larger game.
.45-70 Gummit. With standard pressure ammo it has the trajectory of a basketball and is a very good 100 yrd woods cartridge for big critters. Stuff it with the "magnum" or "+P" loads from Buffalo Bore and Garrett respectively, and you have a decent large/dangerous game combo that crowds the .458 Win. at a very reasonable price. They also flatten the trajectory and extend the practical range a bit.
A hunter from FL just recently took the African Big Six using Garrett loads and a Marlin 1895 rifle. None of the animals he took needed more than two shots.
Yup, I can do it in three. But, for sake of a toy, I'll go with a .............uhhmmmm .35 Whelen Imp. or neck up a .300 WSM to .35 and stick it in a scout rifle. (Hint-Hint Winchester)