Overcoming Evil - Earthbound
Within the game Earthbound, the protagonists are sent on a quest to overcome evil bent on the destruction of the world. This game could easily fit more of the story types, however I feel that it best suits overcoming evil due to its prevalence within the game.
Rags to Riches - GTA 4
In GTA 4, you play as an immigrant that comes to America with the hope of making it big. Going from barely being able to afford rent money to purchasing high-end cars and other vehicles. This is a good example of rags to riches, despite the game title being rather controversial around the topic and others.
The Quest - The Legend of Zelda
A very good and easy to see example. Every LoZ game is layed out in a very similar fashion; the hero, Link, sets out on a quest to rescue the princess Zelda and the city of Hyrule or in some cases, the world. The game follows the quest story very closely, and the character archetypes found also mimic this.
Voyage and Return - Metroid Prime
The reason I picked Metroid Prime for this story type even though the game fits many others is due to the repetition within the trilogy. Samus Aran lands on a planet, voyages through multiple areas and returns to her ship at the end of the game. The games lack an obvious storyline, which allows the players to imagine and immerse themselves within the game world; exploring and creating their own adventures.
Comedy - Simpsons Hit and Run
The game is essentially a parody of the GTA series with humour one would expect from The Simpsons characters and franchise in general. The fact that the game is somewhat a parody easily allows it to fit into this category.
Tragidies - Fallout
The franchise as a whole revolves around the tragidies and effects of nuclear warfare, as the game places you in an environment and setting where you lose your former life/family/companions and venture out into the disastrous wasteland.
Rebirth - Skyrim
Although Skyrim could easily fit into any other category, I feel like the story as a whole fits much more tightly into this category. At the start of the game, you play a prisoner waiting to be executed. When Alduin appears, it is almost as if your character is "rebirth" and given a second chance where you ultimately end up saving Tamriel.