Right off the top, I was going to say a full-auto M-16, but then I thought better of it. In terms of practical shooting, it really doesn't offer much more than the semiautomatic equivalent.
Not the B.A.R. The barrel gets red-hot after firing 1 or 2 full magazines, and there's no way to change it. The magazines themselves hold only 20 rounds. It really needs to be fired from a bipod, and the bipod is an ungainly and floppy thing. Had one, sold it after I got my M-60 (which isn't much to write about either). I do regret selling the B.A.R. because of the investment aspect.
Thompson? It's a great classic, but whatever it does, the M3 Greasegun does better. Again, had one but sold it. I could kick myself simply because they've gone up in value so much.
If I had to pick one full auto, I think it would come down to the Browning M1919A4 belt-fed. Believe it or not, as long as the barrel jacket is not permanently soldered to the trunnion, the gun can be broken down and carried in a briefcase. By substituting the proper parts, it can be changed between .30-06, .308, and 8mm. Or, you can put in an A6 barrel, jacket, and front bearing and make it a bipod-mounted gun. Plus, it can be used on all sorts of vehicle mountings.