While this applies mostly to RPG games (increasing your character's raw power and skills), this may also be a good trait in adventure or story games (in which a character doesn't have any actual skills to raise or make, but through the story (for example, you start as a petty beggar or slave and grow to be a well respected leader or something like that) there's a significant difference between the moment you started the game and the moment you're about to finish it).
Fallout: New Vegas
The main character (the player) starts the game as a sort of courrier, delivering packages across the Mojave wasteland. However, after getting shot in the face because of a package he was carrying, the courrier decides to get revenge on the people who betrayed him. This quest for vengeance ends up getting him knee deep into wasteland politics and climaxes with him having to decide the fate of the entire world.
The player plays as a prince who spends his days in a castle, doing nothing apart from playing with his dog, flirting with his childhood sweetheart and sparing with his mentor. However, when the king attempts to murder him, he flees to the sewers (and into his own mind) where he plots his return to the throne. And before you know it, he's deciding the fate of an entire kingdom, choosing who lives and dies.
Puss in Boots
The titular Puss starts as a frauded thief, who is a somewhat stereotyped "charming rogue". While on his adventure he ends up betrayed once again, then he has a chance to prove to everybody that he is in fact a noble cat and ends up as a savior of a town.
Star Wars: The Force Unleashed
Starkiller is Darth Vader's hitman, obeying his every order and murdering every last surviving Jedi. However, after being betrayed by Vader and realizing that what he is doing may not be morally sound, he decides to turn things around by creating the Galactic Rebellion and confronting Vader.
Every Super Hero Game which involves an Origin Story.
Every super hero was at one point or another, a zero. Spider-Man was an awkward lab geek, Cole was a messenger boy and Iron-Man was an alcoholic, However, through the course of their origin story, they learn to overcome their flaws and become the heroes they were always destined to be.