Before I get to that, though, I'm going to discuss my runners-up. My very first D&D PC was a Cleric, but he didn't last very long. My first D&D PC with any sort of lifespan was a Magic-User who, as I noted before, lived into the late 20s level-wise. That's as far as I've ever gotten a character since- in fact I've not gotten a PC out of the teens in level since then. So I do have a very, very soft spot for spellslingers.
I love Magic-Users, or Mages, Wizards, whatever. Since I cut my teeth on BECMI D&D, I tend to revert to that nomenclature. I wasn't a very athletic kid, so the idea of a bookworm who could potentially become incredibly powerful and wield unearthly forces was very attractive to me. It was also what drove me to learn as much as I could about computers and electronics as a kid, because that's the closest to magic we're going to get in the mundane world. I loved tossing magic missile spells (they never miss!) and phantasmal force and yes, the old favorite, fireball. Magic-User was where it was at, and remains among my first choices when I create a new character for a fantasy game. I'll throw Elf in here, too, as Elves in BECMI D&D are basically multiclass Fighter/Magic-Users. The Elf, as described in Day 2's post, can wear full armor, use any weapon, and still toss spells. This, of course, is limited by slow level advancement- but still. Of course, Elves topped out at 12th level, so they'd never have access to the really powerful magic available to human Magic-Users as they climbed the level ladder. That sort of power took a long time in-game and IRL to amass, but woe to the foe who crosses the mage that wields it. Think fireball is bad? Wait'll you get a load of meteor swarm.
Also in the runner-up category is a class that appears in BECMI only after a Fighter earns name level, a term lost on more recent D&D players that refers generally to 9th level and specifically to the level where a class gains the title by which the PC is considered to have matured in power and standing. At 9th level, a Fighter could opt to become a Paladin, which was a separate class in AD&D. Paladins were warrior knights with a holy devotion to an immortal- like the Crusader knights, but with a bit of clerical magic. Folks that know me generally describe me as having "Lawful Good" ideals, always wanting to do what's right, to help those in need, etc. My friend Scott has referred to it as a "White Knight Complex." Perhaps that's why Paladins appeal to me. This extends even moreso to the Cavalier of AD&D's Unearthed Arcana and especially to the Solamnic Knight of Dragonlance Adventures. The story of Sturm Brightblade, who sought all his life to uphold the honor and ideals of a knighthood that was tarnished and dismissed by society, who succeeded in his actions and in his own death in surpassing those same knights in honor and valor- WOW. What a tale. If Don Quixote de la Mancha had been a younger man, sound of wind and limb, he and Sturm would certainly have had a similar career. The ideals contained in the song "The Impossible Dream" from Man of La Mancha sum up what I love about Paladins. And the world will be better for this- that one man, scorned and covered with scars still strove with his last ounce of courage to reach the unreachable star!
So... this brings me to my absolute favorite PC class. CLERIC. Why Cleric? Well, because of the Elmore artwork in the Expert Set book, initially. MAN those Clerics rocked. Also, Aleena, the Cleric from your first adventure in the D&D Basic Set. But seriously, take a look at the Cleric. Wear any armor. Use any blunt weapon- warhammers and maces rule. Now, gain the ability to make undead creatures run from your holy rebuke. Now, on top of that, add the ability to cast spells that heal and protect, and even a few with offensive capabilities. WOW! That makes for a powerhouse class that amazingly fun to play. You can play the character like the Paladin, just minus the sword. You can play a jolly, Friar Tuck-style Cleric. You can play a vengeful Cleric of an immortal of War, there's so many ways to play a Cleric. All of them, however, are able to mix it up in melee *and* use clerical abilities to defeat undead and heal their allies. Most D&D parties would never leave town without a Cleric. Sure, the Elf can fight and cast- but he can't heal. The only source of magical healing beneath 9th level in BECMI is the Cleric, and it was and remains my favorite D&D character class.