Count me in the camp of those who don't get the point?
There are hundreds of cartridges out there which were not developed for military application, so if you arbitrarily resign yourself to only that list, or even to the list of cartridges which don't even have a military parent or grand parent, you have a lot to choose from. And then you fall into a futile discussion of how many firearms make the minimum battery for hunting, which is a can of worms all unto itself. And then, finally, since you used the word "classic," we fall into yet another layer of pointless debate as to what constitutes a "classic" - which the recent .458win mag post has taught us is nothing more than a subjective bout of conjecture, about as productive as a circle jerk.
For me, a .22 Hornet, .220 Swift, 7mm rem mag, and .416rigby would treat me just fine for rifles. I'd have a hard time with shotguns, I suppose, since 12ga is a military round, as has been 20ga if I recall, and 16ga is just too expensive and hard to come by. Although I do enjoy a SxS 16ga. For handguns, .32H&R, .44rem Mag, and .475linebaugh. Pretty easy, as only the .357mag, .45colt, and pistol cartridges have been military cartridges, and my list certainly should qualify as classics, although two are largely not that popular or common.
So I suppose let the debate begin over which on my list are classic or not. They're all highly effective, and I have killed game with all of them. I don't currently have a .220swift, and ALMOST listed a .22-250 in its place, but since I wanted a .22 hornet on my list for fox and bobcat, I wanted a bit more gap to my higher end 22cal than the .22-250 offered. I ALMOST stuck a .257 Robert in there, but I figured parentage mattered, and I can tone down the 7mm mag enough to cover that gap when needed.