Recurring Joke Joke

A premise for a TV sketch that features several sets of characters who do nothing but re-hash their one joke in different ways. A lot of comedy shows use this technique without relying entirely on it, though there are a few that do, like The Fast Show.

This trope can be employed for several reasons. For example:

Reusing the same characters allows you to subvert the audience's expectations.
It saves money. Imagine how much it would cost to get new costumes, sets and make up for every sketch. Smack the Pony made lots of these dating video sketches which need minimal sets, lighting and costumes this probably helped stretch their budget much further.
It saves time. When you have quite a fast turnover rate, recurring characters are very useful.
You can make very short sketches to break up the pace of the show. The Jesse's Diets bits on The Fast Show were less than a minute long; they could get away with it because the viewer knows the character and knows the format.
It can be good for the show's status. Catch phrases and characters such as "yeah but no but" and "scorchio!" can become institutions. They can grate and become annoying, but there's two sides to every coin.
You can have a narrative within a Sketch Comedy, as in The League of Gentlemen.

Examples of Recurring Joke Joke include:

Film

The Hangover: Part II. The characters repeat all the same jokes as in the first film and the audience knows what's coming all the time.
The Aristocrats. This is literally the entire premise: A bunch of comedians tell the same joke with the same punchline for about 90 minutes.

Live-Action TV

The Fast Show: The entire premise
Little Britain: The entire premise
The State parodied this tendency with the "Louie" sketches, where all he would do was shout his catch phrase, "I Wanna Dip My Balls In It".
Monty Python's Flying Circus had a collection of old men called the Gumbies, who had glasses and hankerchiefs tied in knots on their heads with the recurring joke that they were all very stupid and couldn't talk properly.
- Also the old bearded castaway characters who introduce the show
Saturday Night Live has so many of these that The Other Wiki has started listing them.
Harry Enfield had a few of these, such as the guy who always said, "I don't think you wanted to do that!"
Sonny and Cher
Carol Burnett
Mr. Show did this fairly often, most famously with the old rambling conservative politician character's story about the traveling salesman and the three holes in the wall of the barn where he stays...

Western Animation

The Simpsons, during a Krusty the Clown TV special:

Krusty: And now for our parody of Mad About You, called "Mad About Shoe". (goes to a bed with a giant shoe in it) Hey baby, let's kiss, no tongue. (audience boos) Ugh, You're not going to like our N.Y.P.D Shoe sketch... it's pretty much the same thing.

- Also, Bart invokes this as the "I didn't do it" boy.