Obi-Wan: If you spent as much time practicing your saber techniques as you did your wit, you'd rival Master Yoda as a swordsman. Anakin Skywalker: I thought I already did. Obi-Wan: Only in your mind, my very young apprentice.
Kenichi. Take a guess on how much a teenager can learn about martial arts in 39 volumes and counting.
Bonus points for being an apprentice to multiple masters simultaneously, although they do share the same dojo.
Yahiko in Rurouni Kenshin. He does progress, although rather slowly over the course of the many volumes of the manga and accompanying TV series. At the beginning of the series, Yahiko is most useful to the heroes as a thrown weapon (it happens). At the end of the series he has become a master swordsman in his own right.
The Forwards (Subaru Nakajima, Teana Lanster, Erio Mondial, Caro Ru Lushe) from Magical Girl Lyrical Nanoha StrikerS are the students of Veteran Instructor Nanoha. Other teachers are Vita, Fate Testarossa-Harlaown and Signum, but Nanoha is the main teacher, with Vita as Nanoha's assistant. In Force, they train under Nanoha again.
Elena in The Fairy Godmother by Mercedes Lackey ends up apprenticed to, well, a fairy godmother. Comparatively little of the book is actually devoted to her apprenticeship, however; the plot doesn't really kick in until the job has officially been handed over to her.
In World Without End, Merthin, as an apprentice carpenter, is actually much smarter and more competent than his master and Arch-Enemy Elfric.
Jamie ends up as an apprentice in the Clown Division of The Pilo Family Circus. Because of the mishaps that beset the clowns, he doesn't get to perform much, but he still ends up better off than the previous apprentice...
The protagonist of the Rivers of London series of books, Constable Peter Grant, is a fully qualified policeman as of the first book, but also an apprentice wizard indentured to Detective Chief Inspector Nightingale. There used to be a Wizarding School but the birthrate of people who could use magic dropped making it unsustainable that and most of the people who would've been able to teach magic were killed during World War II.
The Apprentice Rogue: Artamos is a apprentice black knight. This mission is his test to become a full fledged member.
Johnny Tremain in Johnny Tremain by Esther Forbes starts out as an apprentice silversmith.
Maggie Mulligan in Faraway Dream by Jane Flory is an apprentice milliner.
Nabby Jones in Touchmark by Mildred Lawrence eventually becomes an apprentice pewterer.
Menolly is training as an apprentice harper in Anne Mc Caffrey's Dragonsinger, along with numerous other characters.
A Mage's Power: Everyone who joins the Dragon's Lair mercenary company starts off with this rank. Eric is Basilard's apprentice until he proves that he can handle himself in a fight. He's technically a novice for the rest of the book but he's still learning magic from Basilard (and Dengel).
Shintaro Gotou in Kamen Rider OOO becomes one to Akira Date/Kamen Rider Birth, eventually using a second Birth Buster to back Date up. Differs from the Movie War Core continuity, where Date does not appear and Gotou becomes Birth.
In early editions of Dungeons & Dragons it was customary for mages/wizards to learn by becoming apprentices of established mages/wizards. A number of wizard/apprentice relationships appeared in the Forgotten Realms setting. For example, Elminster has had many apprentices over his long life span (hundreds of years).
In Warhammer 40,000 Interrogators are apprentices to Inquisitors. They are slightly more experienced than is usual for the trope; by the time you are even considered for the rank of Interrogator you have probably been serving as a Throne Agent for several decades.
Apprenticeship is the standard way to learn magic in Ars Magica, to the point that a character's apprenticeship (15 years for the default Hermetic magus, and a good few other traditions) is incorporated into character creation. Then again, it's set in 13th-century Europe, so apprenticeship is the standard way to learn a lot of things.
In the Devil Summoner:Raidou Kuzunoha games, Raidou is an apprentice detective (who does nearly all the actual detecting because the detective he's supposed to be learning from is really lazy).
Luke from the Professor Layton series, though interestingly, Luke claims to be his apprentice, and Layton denies it (while letting Luke follow him around and teaching him things).
On Neo Pokeforum Schwarze`s group (Schwarze, N, Molly and Fry) to Janine of Fuschia City, Timber of Cinnabar Island, Silver and Ritchie of Kanto Elite Four, especially the former. The group started as rookies under training of gym leaders and Elite Four members to confront Colress and become professional and high-skilled trainers.
Chowder is one of these to Mung Daal; other child characters are apprenticed to other chefs.
Brainy Smurf is this to Papa Smurf on The Smurfs; likely not the best decision Papa Smurf ever made, and at least on one occasion, he scolds Brainy and says he "has the right mind" to get a new apprentice after Brainy causes a disaster.
Although Terry was technically Batman in Batman Beyond, he could still be considered this to Bruce Wayne. Defied in one episode where Bruce was getting treatment from the Lazarus Pit and the possiblity of him becoming Batman again was brought up; Terry said firmly that he was "not wearing the Robin outfit".
Before the age of streamlined education most (if not all) professions were learned by apprenticing under a master. Children usually apprenticed under their parent until said parent was too old to work, at which point the child would take over the business. This is the origin behind such surnames as Smith, Carpenter, and many other last names that sound like jobs.