During the Logopolis serial we occasionally see this mysterious humanoid dressed in white called The Watcher. We never hear him say anything, and the only explanation we get is at the very end of the story, after the Doctor is mortally injured by falling off the Pharos Project antenna (after saving the universe, of course).
DOCTOR: It's the end. But the moment has been prepared for.
(The Doctor points to a ghostly white figure, that walks forward.)
ADRIC: The Watcher.
(The Watcher merges with the Doctor.)
NYSSA: He was the Doctor all the time.
(Tom Baker turns into the Watcher, then gains youthful features and short straight hair. Peter Davison sits up.)
So...what the heck happened here?
It doesn't make a lot of sense for the Watcher to be a past or future incarnation of the Doctor since he (presumably) ceases to exist after this scene, and he never shows up on the various lists of Doctor incarnations. And how does merging the fatally-damaged Fourth Doctor with some other unspecified "Doctor" thing produce the Fifth Doctor anyway? None of the other regenerations required merging with another entity (I think).
The only explanation I can come up with is that after the Fourth Doctor talked to the Watcher in the first episode of the serial, at some point off-screen he did spacey-wacey things to split himself into two doctors so only half of him would be damaged by the fall, and recombining with the other half would give him enough strength to regenerate. But that's an awfully huge leap. And even that doesn't explain how the Watcher was following them for the whole story.