Doctor Who/Recap/S2/E08 The Chase

Don't just stand there and scream, you little fool, run!

—Ian, to Vicki, speaking for a nation.

Watching the Space/Time Visualiser (a souvenir from last week's "Space Museum"), the Doctor et. al. watch The Beatles, then discover that there's a Dalek time machine on their tail.

In a chase through time and space, the TARDIS with the Daleks in hot pursuit lands on the desert planet Aridius, on top of the Empire State Building, on the deck of the Mary Celeste - the appearance of the Daleks causes the crew to abandon ship - and a fairground haunted house.

We eventually end up on the jungle planet of Mechanus, where a group of robots called Mechonoids take the travellers prisoner. They were sent fifty years earlier to build a city ready for a human colony to arrive and take possession. The colonists never showed up, but the Mechanoids continue to defend the city in readiness. The Doctor and companions meet Steven Taylor, a human astronaut who has been the Mechonoids' prisoner for some time.

The Mechonoids and Daleks battle, mutual destruction ensues, and Ian and Barbara decide to use the rather more reliable Dalek time machine to get home to 1960s London, reluctantly helped by the Doctor. The Mechonoids reappear 40 years later, in the Big Finish Doctor Who audio play The Juggernauts.

Steven Taylor stows away on the TARDIS, and is not discovered by the Doctor until the following serial.

It can be watched here.

It holds the record for most companions in a single episode - not counting the Children of Time during the Tenth Doctor's tenure, but if you count Bret and Canton it shares it with The Dalek's Master Plan and The Impossible Astronaut.

Tropes

Clap Your Hands If You Believe: The Doctor's theory on the (supposedly?) House of Horrors. Although the Daleks clearly believe they're on Earth, their weapons do not seem to incapacitate supposedly "fun-fair" dummies and one of them even throws a Dalek upsidedown and make them run in fear.
Clumsy Copyright Censorship: The Region 1 DVD release has two minutes cut from episode one of "The Chase" in order to avoid paying for the rights to include a short scene with the Beatles. Thankfully, the episode exists uncut in Region 2.
Earn Your Happy Ending: Barbara and Ian finally arrive home after two years. At the time of the show's run, it would be assumed that this is a long time to be time-hopping and that they should have arrived sooner.
Fake Shemp: In a bizarre example of this trope, William Hartnell's regular body double Edmund Warwick was recruited to play the Daleks' android version of the Doctor later in the story. Unfortunately, the director didn't attempt to disguise the fact that Warwick didn't look anything like Hartnell from the front, resulting in an effect somewhat akin to the body doubling from Plan 9 from Outer Space as the android changes from being played by Warwick to being played by Hartnell (and back) from shot to shot.
Ghost Ship: Mary Celeste
Go Mad From the Isolation: Steven.
Go Mad from the Revelation: Apparently, the country boy tourist. Or he just thought the Daleks were completely ridiculous.
Haunted House: Subverted, since it's really a closed funfair exhibit. The Doctor and company never learn this, and neither do the Daleks.
Historical In-Joke: The Mary Celeste, again, as she really existed and really was a Ghost Ship.
Hong Kong Dub: William Hartnell voices the android double of the Doctor even in the scenes where the double is played by Edmund Warwick. Since the BBC's production methods at the time didn't allow for voices to be dubbed on in post-production, this meant that Hartnell had to record the robot's dialogue ahead of time, and the dialogue was dubbed in live during filming. Since Warwick didn't have any reference point for when the dialogue was being played, the dubbing is less than accurate.
Infant Immortality: Shockingly averted. A woman who jumped overboard on the Mary Celeste was carrying a baby.
Just a Stupid Accent: Oh God, the "American" accents. Like it was so hard to hire some barely-professional American extras who happened to be in England at the time. You don't have to be a native speaker to realize they would've been better off to ignore the issues of accents altogether.
- See for yourself!
No Budget: According to the director, the reason why the swamp in episode five is so sub-par is that they had run out of money by that point.
Robot Me: the Daleks' robot double of the Doctor, designed to assassinate the time travelers.
Single Biome Planet: From what we see of both, this trope applies to Aridius (desert planet) and Mechanus (jungle planet).
Stern Chase
Stock Audio Clip: During the Daleks' fight with the Mechanoids, all the lines spoken by the Daleks are from earlier in the serial (in fact, mostly from their first scene in the first episode).
Stock Episode Titles: An odd example, since none of the individual episodes is called "The Chase" - the serial name, however, has 30 uses.
Tonight Someone Dies: "The Death of Doctor Who". Obviously, the robot dies.
Wacky Americans Have Wacky Names: Morton Dill
What Could Have Been: The appearance of The Beatles on the Space-Time Visualiser was originally scripted to be the band playing a concert at some point in the future as old men. The Beatles themselves were reportedly interested, but this was nixed by their management and a promo of "Ticket to Ride" filmed for Top of the Pops was substituted instead. Given what would happen to John Lennon well before he reached old age, this also had the unintended effect of nixing a potential Funny Aneurysm Moment as well.
You Have to Believe Me: Alas poor tourist.
You Look Familiar: Peter Purves plays both Morton Dill (the Southern tourist with the horrible no-good very bad Southern accent) and Steven Taylor, new companion introduced in episode 6 of the story.