My personal favorite is a .280 Rem built on a mauser 98. I honestly don't do anything with it that couldn't be done with a 30-06 or .308 with 150 grain bullets, but I like the weird factor. Had the barreled action I found been chambered in the 1930s Euro 7mm that aproximates the .280 (I forgot the specific caliber designation), I'd be even happier with the same ballistics.
With modern powders and bullets (by modern I mean post 1920), I think there really is a sweet spot for an all purpose medium game rifle from 7-8mm bore x57-63mm with the Mauser basic case. The venerable '06 falls right in the middle of this with a bit more case capacity than the X57mms, making it almost perfect. The .308 comes close, and some of the recent .30's have tried to get to '06 performance with a slightly shorter or blown out case, but they have for good reason not been successful. The '06 just does it well and is trusted to do so. Honorable mention to the .270, but I wouldn't want to hunt moose with it. Some of the niche cartridges do some things better, but there is no better Jack of all trades.
I did have a .31-06 for a little while. It was a real mutt. A Canadian friend had it made up as .311 barrels were easier to find in his part of Canada at the time and the original barrel was junk. Was built on an old Remington (m30?) action. He loaned it to me to work up a load for moose and have a good American gunsmith fix a couple of minor issues. As far as I know, he's still killing moose with IMR 4350 pushing a 174 Hornady RN at a conservative velocity.