Whenever a movie or TV show is scripted, particularly one set in a much earlier time period (or indeed a fictional one, such as Game of Thrones) then the opportunity should be taken for a brief cameo appearance - either entirely in the background, or involving some form of peripheral interaction - by a former or current Doctor and possibly companion(s), and the TARDIS.
As the Musketeers gallop across the grounds of a chateau, the viewer might spot a blue Police box half-visible behind a tree ... in a side booth of the Mos Eisley cantina, two strangers are watching Luke, Obi-Wan and Han with inexplicable curiosity ... There's someone sitting at the table just behind the piano in Rick's Bar in Casablanca who looks oddly familiar...
Basically, Alfred Hitchcock's walk-ons, but with a twist.-- 8th of 7, Jul 01 2017 Chelmsford 123 https://www.youtube...watch?v=NlQyF269spETardis cameo in aforesaid prog. [nineteenthly, Jul 02 2017] Because the TARDIS is connected to all of time and space, wouldn't part of it be at all those places already? The problem would be, spotting where the spacetime fabric stops and the TARDIS starts.
And because Gallifreyans regenerate, the other people in the scene can be timelords.
So this idea is already time travelled.-- wjt, Jul 01 2017 This has occasionally been done, for instance in 'Chelmsford 123'. ISTR, and I suspect you know better than I with this, that it happened once in 'Star Trek' and there were also vague plans to do it in 'Blake's Seven'.
I don't think it would work with GoT because I can't get behind the idea of variable-length seasons (except that I can but that's another story) and I think that places it entirely outside the Whoniverse. If it was like the Helliconia series that would be fine, but it isn't. [+].-- nineteenthly, Jul 02 2017 There's an episode of Midsommer Murders where John Nettles walks into a room and there's an episode of Bergerac on the TV … but that's not quite the same thing, of course.-- 8th of 7, Jul 02 2017 There could be an episode of Dr Who where someone is staring at a computer screen with this idea and this annotation on it, describing them doing this. It's an idea who's time has come +-- xenzag, Jul 04 2017 We need a Dr. McWhoo, where a nearsighted and nearly blind bumbling Dr. stumbles through the universe causing chaos and hilarity while accidentally foiling plots of mire beasts and daleks.-- RayfordSteele, Jul 04 2017 The reverse could be done, in the Doctor Who episodes with scene look-a-likes.-- wjt, Jul 05 2017 random, halfbakery