13th Age does this by starting out with the PCs as notable heroes, they're not recent farmers or wizards apprentices. It's not zero-to-hero, it's hero to greater hero. Not only that, but incremental leveling. Pretty much each session you pick something from the next level: a spell, a feat, hp, etc. After 4 sessions, you level up (with the caveat that the party actually did adventuring stuff/progressed during the session, not just screwed around in the tavern).I already have a narrative challenge with players reaching high levels in short periods of game time.
"Listen here, fledgling 'heroes'! I am a mighty dragon who has spent decades building my power and influence, and putting my plans into motion. You will never stop me! Leave me, and don't come back without treasure I can loot from your corpses."
...a few moments later... "By Tiamat's five gizzards, how in the hell are you so powerful after 1 month!? This makes no sense!"
These days I'm trying out required training downtime/bastion progress to pass time between adventures, but as long as the players feel that there is an ongoing emergent plot, they naturally avoid any non-forced downtime. Can't let the villains rest!
It also helps if you sprinkle in some downtime between adventures.