Clerics, Bards and Druids (be surprise right guys!?) make the ultimate "Party of One". Build versatility as well as access to a wide range of spellcasting both offiensive and defensive.
I'd say short of 5E CoDzilla-ing the game, Paladins make a good 2nd tier Party of One. Again, versatility, lots of good defense and some more limited spellcasting. I'd probably throw Barbarians in there with them. Not much in the way of healing but lots of offense and good defense.
Fighters sound like a good idea for a Party of One, but they're okay. Their offense is moderate and their defense is moderate. If you've eliminated all CoDzilla classes, you're better off going Rogue or Monk and sacrificing your defense for increased offense and mobility.
So: CoD if you want to beat up the game and make it cry, Bards too. Paladins or Barbarians if you want to win but act like you gave the game a fair chance. Wizards can be a great Party of One if they don't have to fight in melee much, which is possible but requires higher teem coordination than the aforementioned classes. Don't be a Sorcerer or a Warlock, be a Wizard. Rogues and Monks before Fighters.
Hey now don't look at me like that! I didn't say Fighters were the worst Party of One.
...I didn't even mention Rangers.