A fan production of a Doctor Who musical prompted this fairly thorough examination of why we haven’t actually had a full-on musical episode of the show, including noting that we have in the audios... and the comics...
It’s a difficult thing to do, and even harder to do well. The Buffy The Vampire Slayer musical episode Once More With Feeling, while not unique in going all-out and having all original songs, still stands out eleven years on - Kate Nash and friends performed it live this Hallowe’en.
It was a singular example of Buffy’s regular-ish format-schmormat episodes.
What similarly singular ideas could work in Doctor Who?
Going past the format-bender and totally beyond the format, for one night only...
We’ve had episodes with the Doctor and companions almost absent, animated specials, the almost-legendary Eastenders crossover which is not canon for either show - no it’s not, la la la not listening...
Century House would have been another example - a show-within-a-show crossover with a to-camera documentary series, two genre-bending notions in one.
A “live” episode with the Doctor and companions saving the day during a special event? This is actually easier to do in an RPG, as it’s “live” by default and you can improvise all kinds of things. By way of example, we’ve had the 2012 Olympics twice now - it would have been thrice but something in the opening ceremony went wibbly-wobbly.
One idea I may have mentioned before, a possible convention adventure avoiding the “who gets to play the Doctor” issue - an alien MacGuffin splits the Doctor into (number of PCs) different aspects, each played by a different British actor, and they have to work together to recombine into one being.
What about a silent episode, like Hush, the flipside of Once More With Feeling? The DWM comics did something like this during Ten and Majenta’s run, visiting a world where a constant psychic broadcast prevents speech and finding it was infested by sound effects that were alive... and hungry. However, in a verbal medium like tabletop RPGs, a silent episode could be difficult - the players and GM can describe everything as normal, so it would have a hazier distinction from a normal session and might be hard to keep up.
One related and possibly more workable suggestion would be to make everyone communicate differently - like the limited vocabularies of cavemen in Robin D. Laws’s RPG Og, as caused by a translation circuit failure.
Consider a non-standard form of storytelling for the show, and acknowledge that it’s unusual.
Like an animated special where someone says “hang on, we’re cartoons!” perhaps caused by being linked to a computer simulation, and the characters use their cartoon-ish-ness to save the day.
Or an old storyteller recounting the legend of the strange travellers who arrived centuries ago in a blue box - and comparing the exaggerated story with the real adventure as you play it out.
Likewise, consider a session using a non-standard rules system.
What would a game be like if you hand everyone a few of RM Bailey’s Doctor Who plot point cards instead of their character sheets?
Or the idea of the storyteller above, based on James Wallis’s adaptation of Baron Munchausen - the game of competitive storytelling, exaggeration and lying? So now everyone is throwing in plot developments which might well get their own characters into more and more trouble...