DCC has a level 0, where you play four terrified peasants with no class and d4 hit points.
Here's the great idea in that: A character should start simple, and get more and more complex as they get less and less likely to die. A first level character can die in a single turn - so you should be able to make one in the same amount of time. As they grow more permanent, leveling up should steadily grow more interesting.
That sounds obvious, but here's the way character complexity grows in most RPG's.
At level 1 you have to roll ability scores and hit points, choose a race and class, buy equipment, calculate all your bonuses, and maybe get an occupation and some spells, skills and feats on top of that.
At level 2, you get more HP and an attack bonus. You may also get a single new spell, or choose one skill to become better at.*
So you spend the most time making decisions about your character at level 1, when they're likely to keel over to a goblin fart ten minutes into the game. Once they have a decent chance of sticking around for a while, you're making a very small amount of choices that have a much lower impact on your character. For instance, a DCC wizard picks 4 new spells at level 1 and 1 new spell at level two. The 0-level helps this problem, but it mostly just postpones that massive complexity mound until level 1.
This is mostly shitty for newcomers, who have to spend a lot of time making a character before they even know how you play D&D. It can reach breaking point if the system is complicated. The first RPG I ever played was 4th Edition, and making characters was just an hour-long slog. Same thing happened with the next RPG I tried, Call of Cthulhu. Both of them have a lot of shit going on at first level, and not much thereafter.
I can understand why things work this way. A character has years of history before you picked them up, and it doesn't make sense for them to suddenly remember their race or occupation at third level. Here's how I'd change things.
Level 1: Roll ability scores and health. Choose race and background.
You roll for a random region, and then choose one of the six races and 6 backgrounds of that region. Each race has a special ability, and determines your starting HP.
The background gives you starting equipment and skills. I like the skills in D&D Next so much that I'd steal the idea for this. For any normal skill check, you roll a d20 and add the relevant ability score modifier. Your occupation gives you four trained skills (e.g. Gather Rumors, Intimidate, Spot, and Use Rope) and you roll an extra dice to add to those skills. This skill dice starts as a d4, and goes up each level.
Race and Background are good starting choices because they rely on real-world logic over game logic. A newcomer might not know what it means to choose +2 AC, but they know how to use a skill called "Intimidate".
So a starting PC has a special racial ability, starting equipment, and four skills. That's a useful, interesting guy.
Level 2: More HP, and Class.
The class you get here would be the simplest possible expression of each archetype. An attack bonus for a fighter, a spell for wizards, a bonus to skills for thieves. Your class is proficient in some weapons.
Level 3: More HP, Specialty, and whatever your class gets when they level up.
Any fiddly extra bits of a class are packed into a specialty. So, a default fighter just gets a boring old attack bonus - at level 3 (when the PC is looking pretty permanent) they get to pick the special cool thing their fighter can do. Falling into a beserker rage, punching through walls, being unstoppable, that kind of thing.
Your class also chooses something every time they level up: a new spell, a new skill, a new weapon to be awesome at, that kind of things.
Level 4+: More HP, and whatever your class gets when they level up. Every three levels, you can choose another specialty.
So hopefully someday soon I'm going to package this and the ability score thing and the summoner class and a random monster generator and everything else into an RPG and release it into the wild.
*Those little lumps in the graph indicate that you'll usually get extra bumps of complexity every few levels. For instance, D&D next gives you an extra feat about every 3 levels.)