Doctor Who
It frustrates me that some of the Doctors are "blocked off" from having new companions in fanfic.
As an example: the 1st doctor always gains a new companion before the previous one leaves the TARDIS, which means that one of the companions in the TV show must [i]always[/i] be there. I can't write a story which is "the First Doctor travels with my fan characters Viola and Vermi and no-one else".
The 1st, 2nd, 3rd, and 5th Doctors are all "blocked off", but...
[continued]
Doctor Who
...each of these Doctors has a different level of blocked-offness.
The 1st Doctor is completely blocked off, unless you set the story on Gallifrey in the time before he stole a TARDIS and ran away with his granddaughter.
The 2nd Doctor is completely blocked off, and in a worse scenario: the 2nd Doctor's travels cannot possibly exceed a few years, because of Jamie, who was with the 2nd Doctor the whole way through... if it wasn't for the workaround (which I'll come back to.)
[cont.]
Doctor Who
The 3rd Doctor is sort of blocked off. He spends most of his time exiled on Earth, which means that most stories are set in the few years that he was working with the Brigadier at UNIT.
But, there is a time jump between Season 10 and Season 11, where:
• he has gained back the ability to use the TARDIS again
• he hasn't got any companions.
You could theoretically put new companions in there, but then you have to explain why he bothers to come back to work with UNIT.
Doctor Who
*There are multiple points in S01E01 where the Doctor takes off in the TARDIS. Which means that from the Doctor's perspective, the story could actually be spread over several months, with him going on a bunch of adventures before returning to London to deal with the Autons.
Doctor Who
With the 2nd Doctor: in his last TV story, he is forced to regenerate by the time lords. But since they hadn't cast Pertwee yet, they don't actually show the regeneration.
This has lead to many stories that star the 2nd Doctor alone, but set while he's trying to run away from the Time Lords so that he doesn't have to regenerate.
Fans call this period "Season 6b", but it isn't actually a fan theory – it's been confirmed as true by the events of The Five Doctors and The Two Doctors.
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The 5th Doctor has a similar situation to the 1st Doctor - there's always an overlapping set of companions, and the Doctor never travels alone.
The 4th, 6th, 7th, 8th, 10th, 11th, and 12th Doctors are all explicitly shown to be traveling on their own.
The 13th Doctor has only had 1 set of companions so far.
The 9th Doctor is a bit tricky - he can never be traveling alone *after* Rose joins the TARDIS in S01E01. But he can be alone before (and during*) that episode.