Bard wins, as usual, though I was originally skeptical (it's easy to get carried by the conceptual inertia of the previous editions). A Lore Bard can ape the best parts of being any other casting class (with the exception of certain subclass features, e.g. the Diviner Wizard's stuff). A Valor Bard gets enough to be decent at most anything (an extra attack, but no further riders; healing, control, damage, and utility spells; some degree of spell "borrowing"). And then all Bards get beaucoup skill stuff: "Jack of All Trades" is literally Remarkable Athlete but for all ability checks, and they double their proficiency on two (and later another two) skills. They've also got Bardic Inspiration, one of the most powerful buffs in the game since it applies to basically everything, and Song of Rest, which is the equivalent of handing out an extra Hit Die (or two!) with a short rest.
Runners-up, as others have mentioned, would be Druid, Wizard, and Cleric, none of which should be a surprise to anyone ever.* Honorable mention goes to the Favored Soul, for creating a (sub)class that can apply Sorcerer metamagic to Cleric buffs and heals--it's not quite a Jack of All Trades, but it's definitely a HUGE flexibility boost to a class that is, very nearly, one-trick-pony (Blowing Stuff Up, that is).
*Well, unless you expected that WotC subsuming the Warlord into the Fighter would mean that a Fighter could actually heal, remove conditions, etc. if built for it. Had the Battlemaster actually been a Warlord, the Fighter might actually have gotten on the "runner-up" list as well. Unfortunately, it isn't, so it's not. (And no, I don't consider "take the Healer feat!" *building* for healing. Because anyone can do that. By that standard, a Chaos Sorcerer is a better support character than a Battlemaster-built-for-supporting!)