Mooching Master

Alice is Bob's master. She could be a master of anything, from martial arts to cooking, anything that involves having a teacher. Bob is very thankful to Alice, since her teaching was invaluable. Perhaps he has more to be thankful for: maybe Alice actually raised Bob, or gave him shelter. Whatever the case, Bob is very thankful to Alice and would do anything for her.

And Alice milks it for all its worth.

Mooching Master refers to a very particular type of teacher-student relationship: the student is indebted to the master and the master takes advantage of it for personal gain. This is not merely the master asking for money, that'd be too simple. The master actually asks for things that go from annoying to illegal to even life-threatening. But the student owes so much to the master, he feels obligated to comply out of a desire to honor her.

This trope is almost always played for laughs, even in a serious work, and the master is not actually malicious (usually...). This doesn't mean her mooching is harmless, though, it can actually make the student get in loads of trouble trying to please his master. And the teacher, while she might be teaching the student with her antics, is, when all's said and done, genuinely taking advantage of him to a degree.

Compare Wax On, Wax Off, where the chores the student is performing are actually genuine teaching tools. Trickster Mentors are very fond of this trope.

Anime and Manga

 * In Ranma One Half, we have ancient grandmaster Happosai, and of course Ranma's own master(and father) Genma. It should be noted, though, that Ranma himself is an equally enormous mooch, so maybe he actually DID learn something?
 * General Cross Marian from D Gray Man abuses his student Allen by making him steal and gamble to pay off all his master's debts, leaving Allen behind to deal with his not-so-friendly creditors, and making Allen do ridiculous things like capturing lions and caring for a man-eating plant.

Film

 * Shop-socky kung fu movies are LOADED with this kind of character. The old Shaolin master ordering his student to get him food and booze is practically a character type by itself. Examples include The Miracle Fighters and Shaolin Vs Lama.
 * Tenacious D in The Pick of Destiny: JB thinks Kage is teaching him how to rock via Wax On, Wax Off, but he's really just cleaning up Kage's house.

Live Action TV

 * Mildred accuses Charlie of being this to Amita, in Numb3rs. She raises a couple of fair points about how he uses Amita's time, attention, research, and resources for FBI work. This leads to Amita carefully distancing herself . . . a little.
 * In Arrested Development, Carl Weathers offers to give Tobias acting lessons, but he ends up talking more about getting free food than actually teaching.
 * House. Give him students and he will send them to get him cable. And donuts.

Video Games

 * Astrid, Rorona's alchemy master from Atelier Rorona. Right at the beginning of the game, her workshop is in danger of being closed and she gets commanded by the king to either do assignments she's requested or forfeit the shop. What does she do? She changes the workshop's name to Rorona's and saddles her with the job. And throughout the game, she's constantly ordering Rorona to do near-suicidal tasks while also treating her like crap, always running from her responsibilities by throwing them on Rorona. Rorona ends in tears constantly at the stuff she does. It's played for laughs, really.
 * In the Raidou Kuzunoha games, Raidou's boss, Narumi, lounges around the headquarters of his detective agency while his apprentice runs all kinds of dangerous errands for him. (He's not a bad guy, though, and does get a bit more involved in the case in the second game.)
 * In the tie-in novel, Raidou Kuzunoha vs. the Dead Messenger, this trait gets greatly exaggerated, as Narumi not only makes Raidou handle agency business by himself but keeps Raidou home from school to do his (Narumi's) housework, spends the money he's given for Raidou's upkeep on fancy new clothes, and threatens to kick Raidou out if he doesn't show proper gratitude.

Western Animation

 * A couple of Looney Tunes shorts feature a fat, lazy cat named Dodsworth, who passes himself off as a teacher and has his young pupil do all his mouse/bird catching for him.