It seems like the closest that we will be getting to Portal 3 is through a mod for Portal 2. Portal Stories: Mel is a mod created by eight Portal fans after three years of development. The mod itself has such a large file that it has to have an individual release and get approved through Steam Greenlight.
Luckily for us, it was approved on Greenlight and is set to release on March 31st, 2015. It's free-to-play so all you need to play this is a copy of Portal 2.
Big file means big mod, right? Yes, yes it does. The mod will have you playing through an entirely new story, with over 20 puzzles and maps. You will play as Mel in the early years of Aperture, your traveling partner will be a new personality core and your Portal Gun will be a 70's prototype.
There's plenty of story to around for this mod, be sure to check it out on Steam.