1. Ace
Ace was not your average companion. Short-tempered, afflicted with self-doubt, and fond of explosives, she frequently exasperated the Doctor. Still, the Doctor realized her potential even if she didn't, and he enjoyed acting as her mentor. Ace, in turn, benefited from the Doctor's centuries of experience and gained self-assurance she wouldn't have found on her own. When Doctor Who ended its first run in 1989, it was clear that Ace was going to keep growing and changing...but into what? Although several audio adventures and novels have served to continue her TARDIS travels, we've never learned anything specific. And at some point she and the Doctor parted ways because she was nowhere to be seen in the 1996 TV movie, but how did she leave? Did she tire of the excitement of time and space and decide to return to boring old Perivale? Did she decide to make a new home as far away from Earth as possible? Or did she die in some space battle, lobbing cans of Nitro-9 as she fell? If she is still alive out there, I think the Doctor would enjoy a chance to catch up with his old student and see just how far his tutelage took her.