Generally speaking, any given Doctor Who episode may include four different kinds of characters. You have your main heroes (the Doctor and his companion(s)), your episodic villain, your episodic heroes, and your background characters. The most intriguing of these is the episodic hero, an optional role that usually consists of a character who impresses the Doctor and plays an integral role in stopping the villain, often by coming up with a brilliant idea or sacrificing themselves to save others.
In many cases, a new companion starts out as an episodic hero, but they happen to be around when the Doctor is alone, which grants them a spot in the TARDIS. But throughout the show, there have been a few episodic characters that clearly felt like they were supposed to be companions, but they end up either dying or being left behind instead.
Donna Noble is one of the best examples, because she wasn't intended to be a companion when "The Runaway Bride" first aired. However, the fandom liked her and the cast and crew liked Catherine Tate, so she was brought back for Series 4. Many other characters weren't quite so lucky, but they stood out to fans as far more memorable than the other episodic characters.
One of the prevailing theories about this upcoming season is that the Big Bad will actually be a character that the Doctor left behind, someone who is bitter enough to scheme against him and the people who were allowed to travel with him. Following that track, I believe that "The Robot Revolution" started a trend where every episode this season will introduce a character that, by all rights, should have become a companion, but doesn't.
A brief exploration of would-be companions over the last 20 years
Since Doctor Who came back in 2005, there have been plenty of characters who would make good companions, if they were given the opportunity. Series 1 actually introduced a character that got to briefly become a companion only to be abandoned for his own failings (Adam Mitchell), but those kinds of situation are far outnumbered by the people that would've been great but never got the chance.
Some of these characters are ones that fans enjoyed, but there wasn't any real need for them to come aboard. For example, Sally Sparrow was a great protagonist in the Doctor-lite story "Blink", but the Doctor wasn't looking for a new companion and thus she never appeared again.
However, there were also several characters that were practically guaranteed a spot on the TARDIS, but who didn't actually get to travel with the Doctor. One of the earliest examples was Lynda Moss, who got permission to join the Doctor and Rose but died on Satellite Five instead. Other characters offered this chance included Madame de Pompadour, Astrid Peth, Rita ("The God Complex"), and Osgood.
Finally, there were some characters who wanted to travel with the Doctor but were rejected instead. One of the cruelest examples of this was in the Series 8 episode "Into the Dalek," when the character Journey chased after the Doctor and Clara, asking to join them, but was turned away because she was a soldier.
There were plenty of characters who would have made good companions, but they either never caught the Doctor's interest or died before they could join him. While the show has addressed the Doctor's guilt over those who died, it hasn't done much to explore those who were just left behind. And that gap is precisely what I think Season 2 is going to be all about.
Two episodes, three possible companions
While "Joy to the World" wasn't technically part of Season 2, Doctor Who Christmas specials typically help set up the themes and plot arcs of the upcoming season. And it certainly focused on the people that the Doctor leaves behind. Despite Ruby's departure being amicable, the Doctor refusesd to even call her because he insisted on moving forward alone. But he also met two other characters that had the potential to be great companions.
The obvious one is Joy, who the promotional materials spent a lot of time on. She went on a crazy adventure with the Doctor, but she also had the kind of emotional depth that a companion needs. The Doctor described her smile as "the lid on a boiling pot," drawing out her anger and pain from her mother's death. Perhaps she would have become a companion, but she died for the star to rise.
But like many would-be companions, Joy told the Doctor to find a friend, to gain a companion to help him. Yet he didn't turn to the obvious candidate—a woman who he'd spent a year with, one who was clever and interesting and devoted to him. Regardless of the episode downplaying her, many fans saw Anita as the one who should be the Doctor's next companion.
This is a core element of the theory that Anita is actually Mrs. Flood, because she had an incredibly strong bond with the Doctor. They spent a full year together, fostering a literal sense of companionship, until the Doctor abruptly left her to finish his adventure. Like so many would-be companions, she seemed to only exist to get the Doctor ready for the new one.
But why wasn't she just made a companion outright? After all, the Doctor apparently thought highly enough of her to recommend her for a different time travel-related position. In the real world, the simple answer is likely that there was already a companion planned, and Steven Moffat's one-off characters were never going to become Davies-era companions. But there might be another explanation.
It was odd to have two possible companions in the Christmas special, but for neither of them to claim the role. But then the first episode of the season proper released... and there was another possible companion! "The Robot Revolution" introduced new companion Belinda Chandra, but it also introduced Sasha 55.
Sasha 55 was a citizen of Missbelindachandra One who spent six months with the Doctor, planning and fighting together. She told Belinda that she'd learned to trust him with her life and told him to "take me to the stars." And then she died. Yet another would-be companion depicted, and another would-be companion gone.
Will every episode in Season 2 feature a would-be companion?
As stated earlier, it's not unusual for Doctor Who to introduce characters that feel like they should become companions, only to leave them behind or kill them off. The Doctor Who website even posted an article on the subject in February! But it is very strange to have it happen three times in a row. Anything could be a coincidence, but it seems very likely that this is a sign, not an accident.
I believe this is the start of a trend. Every episode this season will include a character who, in another circumstance, would become a companion. But for some reason or another, none of them will make it onto the TARDIS, prompting Belinda to doubt the Doctor even more. When he offered her all of time and space, she told him that he was dangerous, promising impressionable young women the world, only to get them killed.
If I'm right, that pattern will continue, with Belinda constantly torn between trusting the Doctor and being disgusted by him. Then the first part of the finale will introduce a villain who speaks for everyone who was left behind. That might be Mrs. Flood, or it might be someone like Ashildr/Me from Series 9. But the first half will end with them inviting Belinda to join them.
The finale (and thus the fate of the universe) will then hinge on what Belinda decides. It seems likely that she will join the villain for a time, seeing the Doctor as precisely the same kind of condescending jerk that Alan was. Ruby will probably play the opposite role, sticking with the Doctor until the climax, when either she or Belinda convinces the other to switch sides. Which way that conflict goes will almost certainly depend on how other companions and would-be companions are treated.
Not everyone can be a companion, but it's shocking that the series hasn't really bothered with those who lost their chance before now. The episode "The Power of the Doctor" addressed the psychological damage it did to the companions to be whisked away for years, just to be dropped back in their day-to-day life without anyone to share it with. For many former companions, there was a real sense of abandonment involved.
But just think about the impacts on those who didn't even have the chance to see the universe. Most of them met the Doctor on the worst day of their life, but they also learned that it was possible to travel in time and space, an adventure that would be worth the danger. But they never actually got the chance to experience it! There's something remarkably cold about that, and this season seems poised to finally address it.