1. The Power of the Daleks
This had to be number 1 on the list. I've enjoyed the animations and even the photo-reconstructions. But The Power of the Daleks is still top of the list of stories that I hope they recover in full.
This is partly due to how significant the story is. The first ever "new Doctor/post-regeneration" story is incredibly important to the show's history. If this didn't work, Doctor Who would've likely ended not long after. Instead of a TV legend that's lasted over six decades, it would've simply been a fascinating sci-fi curiosity, one that worked for a while until they jumped the shark by replacing the lead actor.
So it's an enormous credit to the production team that they did find a way of making the change work. Bringing back the show's biggest monsters for the new Doctor's first story probably helped to ease that transition. Along with the companions Ben and Polly, seeing the Daleks again could've helped to bring a sense of familiarity to the audience.
Having watched all the surviving episodes of Patrick Troughton's era, one thing that leaps out is how physical his performance is. In every story, he brings a lot of energy to the role, and it's easy to see why Matt Smith was so directly inspired by Troughton's performance when playing Eleven.
It's also another sad reminder of just how great a loss Troughton's missing episodes really are. I'd honestly love to see all of his stories restored in full (yes, even The Space Pirates). But if there's one single story I'd love to see more than any other, it has to be his debut story. Getting to see how he plays the Doctor for the first time, especially with the almost impossible task of taking over from William Hartnell, would be amazing to watch.
It's almost doubtful that any of these missing episodes will be found. But considering how many episodes have been returned to the BBC Archive over the years, we can continue to live in hope that more will be found and seen by a brand new audience.