Soldier (d20 Modern)
Note: Technically an Advanced Class.
What It Offered: Its Tactical Aid class feature proved great fun for a character focused on combat but with a secondary schtick of party buffing. Although, by later levels, that character started to *focus* on party buffing courtesy of this class.
Could It Have Been Done w/o the Class: Non-magical party buffs? Not at that time. This was before the MiniHB, so the Marshal class wasn't available. Int-based party buffs? Still not available anywhere else, to the best of my knowledge.
Frenzied Berserker (Complete Warrior)
What It Offered: A unique 'boss' encounter that required the PCs to think outside the box and deal with an enemy they couldn't kill using normal means, without moving away from human foes.
Could It Have Been Done w/o the Class: With some handwaving or extensive templating, yes. The latter would have had a very different vibe, however, and wouldn't have fit the campaign it was used in.
Shadowbane Inquisitor (Complete Adventurer)
What It Offered: The single most badass PC I've seen in any game.

I was the GM in this game and by the end, practically every NPC and PC, including ones who could have taken this guy out in combat, were in both awe and terror of him and the players and myself were loving every minute of it.
Could It Have Been Done w/o the Class: The roleplaying could have, but even in a 'no alignment' game I would have had to strip him of his paladin status if the code of conduct was to mean anything. Also, I'm given to understand the class inspired the character.
Mystic Theurge, Arcane Trickster and Eldritch Knight (3.5 Dungeon Master's Guide)
What They Offered: Functional caster multiclassing.
Could It Have Been Done w/o the Classes: Only with houserules.
Martial Arts Master (Blood and Fists)
Note: Technically an Advanced Class.
What It Offered: The wicked cool Blood and Fists martial arts system. Lots of NPCs and a few PCs, including basically every scro, have had a level or two at least.
Could It Have Been Done w/o the Class: I suppose it could be rewritten to work without the class, but it's a perfectly functional mechanism for the system, so why?