September 1: The Pirate Planet Parts Three & Four

Of course, the Mentiads are on the Doctor's side, so it turns out he wasn't in any real danger.  This means we get a chance to get a bunch of exposition about what's going on: why the Mentiads grow in strength every time Zanak plunders another world, and who Queen Xanxia was and why the Captain was able to take over so easily (basically, Xanxia left Zanak in ruins and easy to conquer).  And there's the memorable scene inside the Captain's trophy room, as he explains to the Doctor how he's created a feat of gravitational perfection by having all the superdense remains balanced against each other, so that they don't immediately collapse into a black hole.

DOCTOR: ...it's the most brilliant piece of astro-gravitational engineering I've ever seen.  The concept is simply staggering.  Pointless, but staggering.
CAPTAIN: I'm gratified that you appreciate it.
DOCTOR: Appreciate it?  Appreciate it?!  What, you commit mass destruction and murder on a scale that's almost inconceivable and you ask me to appreciate it?  Just because you happen to have made a brilliantly conceived toy out of the mummified remains of planets–
CAPTAIN: Devil storms, Doctor!  It is not a toy!
DOCTOR: Then what's it for?!  Huh?  What are you doing?  What could possibly be worth all this?

(from The Doctor Who Transcripts - The Pirate Planet)

It's a fabulous performance from Tom Baker, who usually portrays the Doctor as in command of the situation and largely unaffected by events around him (see, for instance, his behavior not long before this scene, where he's chained up to a pillar but still manages to be in complete control: "You don't want to take over the universe, do you?  No.  You wouldn't know what to do with it.  Beyond shout at it") but here, his sheer horror at the situation bubbles over into the shocked outrage we see here.

The Nurse, the Captain, and Mr. Fibuli look as the Doctor produces
the destroyed Polyphase Avatron. (The Pirate Planet Part Three) ©BBC

And part three also features the fight between K-9 and the Captain's Polyphase Avatron (aka robot parrot), in a bizarre yet entertaining fight.  K-9 keeps blasting up while the Avatron flies around pooping death on everything.  Of course K-9 wins, but it's still entertaining to watch.  Plus it leads to the discovery by the Doctor and Kimus of Queen Xanxia, trapped in her last few seconds of life while giant time dams have been constructed to prolong those seconds as long as possible.

But as good as part three is, it's really the cliffhanger that shines, both in its execution and in its resolution in part four, as we learn that the Doctor has not actually walked the plank to his death, but rather a projection.  This little bit of trickery changes what we thought we knew about the motivations: we thought the Captain was power-mad for some reason, but it's actually his nurse (who didn't even appear in part one) who's been pulling the strings: she's a projection of Queen Xanxia, attempting to rule Zanak forever.  Everything after this just involves trying to stop her -- entertaining to watch (and touching too, in the case of the Captain's reaction to Mr. Fibuli's death), but there are no new twists in the tale.  The Doctor is able to stop Zanak from moving on and Xanxia is defeated.  Oh, and it turns out that the entire planet of Calufrax was the second segment of the Key to Time, which they apparently recover from the vortex.

It's an immensely clever tale, with quite a lot of plot running through these four episodes.  This does occasionally mean that The Pirate Planet has exposition scenes where the Doctor and Romana have to explain the plot to the Mentiads/Kimus/Mula/each other, but honestly it's difficult to mind much.  The inventiveness on display, combined with the coating of fun that these episodes have, makes The Pirate Planet a very satisfying story to watch.