This video really is Doctor Who: Every Story 1963-2013, as compiled by the good folks at BabelColour. 13 Doctors in 13 minutes! And they don’t (he doesn’t would be more accurate, for even if his videos are group efforts, there is only one mind driving these productions) just cover the primary TV series; they also include the spinoffs, the specials, the webisodes, the games, and so much more. There is logic in the things he does not include as well; enjoy.
UPDATE: As some folks were kind enough to point out to me (see the comments), this is not a group effort, but the work of one person, named Stuart. That definitely makes this even more impressive than I thought it was when I believed there was an entire production team behind this video. The flip side of that is, I could have sworn the credits at the end of the video acknowledged other people who contributed to the production, several doing voice over work, one creating the music mix, and so forth. I suspect this may be a difference in degree, not kind, since the director always has the vision of what the final product will be, and assembles and edits the various parts of the production team’s work into that final result, after assigning them each their tasks and explaining what he/she needs from them.