Dawn, Buffy's sister in Buffy. She gets in trouble so frequently that it gets a Lampshade Hanging, and gets all huffy whenever someone calls her a kid. Or, at points in the sixth season, for saying practically anything at all. It gets to the point where Dark Willow threatens to kill her just to put an end to her constant whining. She's grown out of it by Season 7.
Connor from Angel takes this trope Up to Eleven. He's like Dawn only even more Wangsty and with super-powers, allowing him to interact with the plot all the more.
Max from Power Rangers Wild Force who is in love best friends with Danny and thus hangs off him most of the time. While he doesn't suffer from needing rescues very often (he is a Power Ranger), it doesn't help that one of the two girls on the team is ex-Air Force and second-in-command, while The Chick is a responsible college student that was trained from childhood in martial arts by her sensei father, and thus both are far more capable in battle than he'll ever be.
Drake & Josh has Megan, who is a young evil genius that frequently gets the boys in trouble.
Damon, from Heroes. He toes the line when after Micah tells him that, even though his vintage comics from his late father are worth a lot of money, he still wouldn't sell them. Damon steals them from Micah and not only sells them himself, but also sells the medal of valor that was awarded to Micah's dad for saving a little girl from a fire and is the only memento Micah has of his father. When Micah confronts him about this, Damon tells him, "I'll buy you a new one."
Kenard from The Wire. What separates him from others is that he is really psychotic, putting lighter fluid on a cat in one scene and committing a murder in the same episode.
Lilith from Maddigan's Quest has Lilith, who verges on being the show's Damsel Scrappy. Her attention whoring almost gets Eden killed in Witchfinder, whilst in other episodes she does little more than get herself into trouble and whine at the people causing it until she's rescued, and despite the fact that the show's timeline undergoes two major changes in-series, she doesn't change at all.
Nog in the earlier seasons of Star Trek: Deep Space Nine. This is partly excused by the characters as a part of his Ferengi nature - except that Rom and Quark seem to get annoyed with him as much as anyone.
Mitch's son Hobie on Baywatch. A regular 'victim of the week', constantly almost drowning despite being the son of a lifeguard.