List of Quests with a level cap.

Lazy update:

The Story Behind the Case
Exploring Mushroom Forest (1)
Exploring Mushroom Forest (2)
Exploring Mushroom Forest (3)
Friendship with Bruce
Killer Mushroom Spores (1)
Killer Mushroom Spores (2)
Over the Castle Walls (1)
Over the Castle Walls (2)
Over the Castle Walls (3)
Over the Castle Walls (4)
James's Whereabouts (1)
James's Whereabouts (2)
James's Whereabouts (3)
Stop the Wedding
Where's Violetta?
The Identity of the Princess
The Seal of the Empire
The Story of Betrayal
Truth Revealed
Eliminating the Rest
The Recovered Royal Seal
Sending Provisions (1)
Sending Provisions (2)
King Pepe's Scroll

I might have missed one or two.

Edit: Since there's this new <Job> Training Center for all Adventurers under level 20, I thought I should include this:

Training Center for Beginner Warriors
The Lost Letter of Commendation
Regaining Confidence
Stump's Weakness
Using Power Strike
Team Training is Fun! (Party)

Please note that these are for warriors, although the names are probably not that different for all other classes.