Actor Allusion: This has happened with numerous guest hosts and panellists to different extents:
When Bruce Forsyth was a guest presenter for the first time, the show turned into a parody of Bruce Forsyth's Generation Game and Play Your Cards Right, the latter as "Play Your Iraqi Cards Right", a reference to the "deck of death" to the utter bewilderment of Hislop and utter delight of Merton. His second appearance featured incidental music from Strictly Come Dancing, and ended with the points revealed in the style of that show.
Hislop later described "Iraqi Cards" as being quite possibly the most tasteless thing the show had ever done. (It could have been worse — the writers wanted to call it "Play Your Kurds Right".)
Jerry Springer's episode featured descriptive captions for the panellists and his trademark 'final thought'. Also, Paul suggesting they have a lie detector test, and Ian suggesting a fight.
William Shatner opened the show with the background sliding open like automatic doors with a familiar ssshtk sound, and the host's chair had been modified to resemble the captain's chair on the Enterprise. Then there were lots of little phrases hidden throughout the show relating to Star Trek. He even did a song round, reference to his singing career. Also all the guests did their best to bring up Star Trek at every opportunity.
Damian Lewis: What's Radio Four for? Paul: Radio 4-4? Is there two of them? [...] Is there a Radio 2 too? Harry Shearer: Radio 1-1 makes 11. Damian:"Look it goes to eleven." Harry: Ohh, yeah. Audience: Ahhh. Damian: Spinal Tap joke! Harry:I knew it would happen. Damian: Got there early. I am so, SO happy I've got that in.
Colbert Bump: Boris Johnson is the biggest example — he became famous enough to become Mayor of London because of this show. Lampshaded by Paul Merton frequently, remarking "His plan worked, so he won't be coming back". Other examples have followed to a smaller extent, as seen below.
As Dara O Briain puts it: "If I knew that's what you get from hosting Have I Got News For You, I wouldn't have settled for Mock the Week."
Many people have appeared as a panellist before being upgraded to the host's chair.
For a non-person example, one episode featured a signed photograph of Ian Hislop on eBay that had attracted six hits and no bids. One week later, it had attracted a maximum bid of over £110.
Politicians sometimes take a gamble to try and get the same success Boris has. Unlike him, however, they are often not that successful. Among the more successful would be William Hague, former Conservative Party leader, who ended up doing a few years as a rather successful after-dinner speaker before eventually becoming Foreign Secretary in 2010.
Some writers have tried using the show to springboard their books being launched. Also unlike Boris, it tends not to work for varying reasons.
Direct-to-Video: Two special episodes in the 90s, Have I Got Unbroadcastable News for You and The Official Pirate Video.
Miles Jupp's appearance in series 40 successfully got "Balamory" trending on Twitter in the UK.
Internet Counterattack: There was a time when "conniving little shit" was a redirect to "Rupert Allason" on The Other Wiki. The redirect is gone, but he's still the top search result.
Keep Circulating the Tapes: As it is a topical programme, no full releases of series have been made. Many episodes from across the series were available on YouTube until a spate of copyright claims put an end to that. Other copies do exist.
Peter Hitchens, who appeared in an episode in series 17, afterwards described HIGNFY as being a "repellent programme".
The appearances of Jimmy Saville and Max Clifford are this for the staff and panel.
One of Us: Paul wore a The Prisoner jacket in one early episode that aired when Channel4 were beginning to repeat the series.
Playing Against Type: Intentionally done with several of the guest hosts, as people such as Boris Johnson are placed in a position where they are expected to keep order and move the show along smoothly.
Promoted Fanboy: Bruce Forsyth wrote to Paul Merton asking if he could host one of the shows, saying he really enjoyed them. Sure enough...
Screwed by the Network: The 39th series, shown in Spring 2010, was moved out of the Friday slot the show had occupied for nineteen years to a Thursday slot. The reason given for this was so they could show an Election special on May the 6th, the day Britain went to the polls. However, it then turned out that if the Thursday broadcast went ahead, the show would not be allowed to discuss the election at all in case they influenced the way people voted (as it would be broadcast before polls had closed). The episode was hence hastily rescheduled... for the day after, which was of course a Friday. Quite why the BBC thought they could put a satirical news programme on before the polls had closed on election day under broadcasting regulations is beyond anyone.
Too Soon: Especially the show which aired immediately after the BBC issued strict instructions that no programme was to raise the matter of Peter Mandelson's private life in any way...
Immediately followed by Merton — "What's wrong with gay people owning homes?"
The News Quiz, a radio programme from which Have I Got News for You was originally adapted, did a similar gag stating that "We are unable to say that (classical music fragment) is (another classical music fragment). As the BBC were very much locking the stable door after the horse had bolted (and possibly mounted another horse along the way), everyone knew exactly what was being said.
Hislop also managed to call Mandelson a "friend of John Birt" (the BBC boss who made the decision) in a way that suggested it was an Unusual Euphemism.
Writer Revolt: When Merton was teamed with disgraced ex-MI-5 agent David Shayler displayed on a television satellite feed with a two-second delay, he refused to play along for the first half of the show (due to Merton's strong belief that timing is the most important part of comedy, which a two-second delay greatly interfered with) and even turned the television off at one point.
Stephen Fry treated said ex-agent quite viciously as well, beginning by referring to him as "David Traitor" and going from there.