Doctor Who/Recap/S10/E01 The Three Doctors

Two Doctors are better than one. (When they aren't bickering, that is.)

So you're my replacements - a dandy and a clown!

—The First Doctor

Third Doctor: Well Sergeant, aren't you going to say that it's bigger on the inside than it is on the outside - everybody else does.

Sergeant Benton: ... it's ... pretty obvious isn't it?

The tenth anniversary episode of Doctor Who was marked with this, the first "Multi-Doctor" episode, a concept which would be used several more times over the coming years.

The Doctor is investigating the appearance of a mysterious blob-like creature which, it transpires, is hunting him down. Reluctantly, he calls on the Time Lords for help. Unfortunately, they're a bit preoccupied with the black hole draining all their energy, so they send his past selves to help. The First Doctor is caught in a time eddy and can only advise via the TARDIS viewscreen, but he deduces that the blob creature is a "time bridge". The Third Doctor and Jo allow it to capture them and they are transported to a world of antimatter beyond the troublesome black hole.

Meanwhile, back on Earth, the Second Doctor, the Brigadier and Sergeant Benton take refuge in the TARDIS. The First Doctor tells him to switch off the forcefield and the TARDIS (and with it the UNIT HQ) are all transported to the antimatter world.

It turns out that the problem is Omega, a renegade Time Lord. Omega was a stellar engineer, and his experiments enabled the Time Lords to develop time travel. But one of his experiments went wrong and trapped him in the antimatter universe. Now he wants to return, but needs a matter counterpart to take his place, and he wants the Doctor.

Driven mad by his isolation and corroded so that only his will survives, he threatens to destroy the universe, but he is tricked into picking up the Second Doctor's recorder - the only thing not converted to antimatter when they travelled through the black hole - and is destroyed.

As a reward for saving the universe, The Doctor's exile is lifted and he is able to use his TARDIS normally once more.

Tropes

Arbitrary Skepticism - The Brig is rather accepting of the whole anti-matter monsters and black holes situation (he's dealt with weirder, arguably). But he immediately dismisses the idea of two Doctors existing at the same time (he thinks Three must have changed back somehow), and insists the TARDIS interior must be an optical illusion.
A God Am I - Omega
Battle in the Center of the Mind - The Third Doctor challenges Omega to one of these... and loses, badly. The only reason he doesn't die is because the Second Doctor talks Omega out of it.
Big No - Omega pulls this one off in a disturbingly serious way.
Chekhov's Gun: The Second Doctor's recorder.
Exact Words: The Second and Third Doctors pledge that they will not attempt to leave Omega's realm before Omega himself. Omega assumes this means they'll stay with him forever. It doesn't.
Future Me Scares Me - Or rather Future Me Disappoints Me. Also inverted, since in this case Future Me Isn't Exactly Impressed With Me Either.
Large Ham - Omega.
- Everybody, Everybody gets at least one taste of pork products and/or scenery during this serial (for instance, ask The Brigadier his opinion on the Second Doctor's recorder...). That said, Omega. It's a wonder they were able to find any more ham for anyone else once Omega had gotten started.
Milestone Celebration
Morton's Fork: Omega can only escape from his realm by relinquishing control of the Singularity he uses to create the realm. Unfortunately, the very moment he relinquishes control of the Singularity, everything falls apart—preventing him from escaping.
The Nth Doctor
Obfuscating Stupidity: At one point, the Second Doctor rambles to Omega about his lost recorder, prompting (yet another) tantrum on Omega's part. The Third Doctor reprimands him, only for the Second Doctor to reveal he's just been testing Omega's self-control to find a weakness.
Omnicidal Maniac - Omega could give Davros a run for his money. See Villainous Breakdown.
Percussive Maintenance: Radio not working? Smack it on the TARDIS console. Voila!
The Rival: Doctors Two and Three took an immediate dislike to each other, and began sniping at each other almost from first meeting themselves. Patrick Troughton and Jon Pertwee, in Real Life, loved the effect so much that they decided to carry on the mock bickering, in character and out of character, Jack Benny and Fred Allen-style, in public for the rest of their careers (in private, they actually became quite good friends).
Real Life Writes the Plot - William Hartnell was ill and getting on in years (in fact this serial was the last time he acted before his death), which is why the First Doctor appears only on a viewscreen.
The Scream - Omega lets out a rather convincing primal wail when he discovers that there's literally nothing left of him but his mind and some dust, animated by willpower alone.
Villainous Breakdown - Omega goes completely off the deep end when he finds the aforementioned existence failure. "If all that is left of me is my will... Then my will is to DESTROY ALL THINGS! AAAALL THINGS! ALLLLL THIIIINGS!!! *cue second anguished wail*"
Waxing Lyrical: "I am he and he is me." "And we are all together, goo-goo-kachoo?"
You Look Familiar - The Chancellor of the Time Lords is played by the same actor as one of the Time Lord judges who exiled the Doctor to Earth in "The War Games". This was deliberate on the part of the production staff, in order to create a link back to The War Games.