Currently in spot 5 of What's reddit playing!
Ok, while this is not exactly a game per se it does involve choices and it does have a storyline. It is a visual novel called Katawa Shoujo. Here is the information from the "About" section:
Katawa Shoujo is a bishoujo-style visual novel set in the fictional Yamaku High School for disabled children, located somewhere in modern Japan. Hisao Nakai, a normal boy living a normal life, has his life turned upside down when a congenital heart defect forces him to move to a new school after a long hospitalization. Despite his difficulties, Hisao is able to find friends—and perhaps love, if he plays his cards right. There are five main paths corresponding to the 5 main female characters, each path following the storyline pertaining to that character.
The story is told through the perspective of the main character, using a first person narrative.
Really gets you wrapped up in the story and to be brutally honest I *may* have broken down once or twice. Each path is around 5-ish hours.
Yes it is animated in the "Anime" style but the story is really worth it, and it's free!
Works on Windows, Mac, and Linux!
Here is the site: http://katawa-shoujo.com/
NOTE: There are some scenes with "adult content" but they can be toggled.
Post your favorite character! Mine is Hanako (as seen in my sig.).