Constrained Writing

This is when an author writes in an atypical pattern. The reasons for this can vary, from Leaning on the Fourth Wall (if it's related to the story in some way), to keeping certain plot points and twists hidden to the very end (e.g. avoiding gender pronouns for a Samus Is a Girl twist) to simply being a stylistic choice. Some types of self imposed challenges can include writing in a particular metre (e.g. using a fixed pattern of syllables or making each line a letter longer than the one that preceded it), writing in code (e.g. replacing words with ones that appear a few places afterwards in the dictionary) avoiding certain common letters (the correct term for this is a lipogram, by the way) and words or displaying some sort of complex pattern (e.g. making large chunks of the story alliterative or in palindromes), drabbles (stories of precisely 100 words) and many more.

Remember this applies to any challenge imposed on the author by themselves, so normal deadlines and schedules don't count, however improvising with limited resources or using a particularly strict time limit does.

The alternative name "Oulipo" is from a group of French writers who were dedicated to this style of writing. The Other Wiki also has an article on the concept.

Some authors might adopt this as a Signature Style. Unconventional Formatting can be related. When this is used within an otherwise normal piece of writing to show something specific then it is Painting the Fourth Wall.

Examples of Constrained Writing include:

Web Comics

xkcd has a comic about this.

Western Animation

The Simpsons:
- When Homer becomes a food critic, he writes a review without using the letter E. That's because the E key on his typewriter is broken.
- Mr Burns tells Lenny that he will be fired unless he is able to explain why he shouldn't be without using the letter E.

Lenny: Uh, I'm a... good... work... guy...
Burns: You're fired.
Lenny: But I didn't say...
Burns: You will. [pushes trapdoor button]
Lenny [falling]: EEEEEEEEEeeeeeeeeeeeee!

Real Life

A Void (and La Disparition, the French novel it's a translation of) was written without using the letter E.
Gadsby: A Story of Over 50000 Words Without Using the Letter "E" by Ernest Vincent Wright.
Mike Keith has rewritten Edgar Allan Poe's The Raven with the limitation that words must have the same number of letters as the corresponding digit of pi. Read it full here: http://www.cadaeic.net/naraven.htm
Le Train De Nulle Part is a French novel with no verbs.
In her first Thieves' World short story, "The Secret of the Blue Star", Marion Zimmer Bradley carefully tried to avoid referring to the gender of the magician Lythande to conceal the Twist Ending that Lythande is a woman. She did slip up at one point, however.

Lythande drew from the folds of his robe a small pouch containing a quantity of sweet-smelling herbs, rolled them into a blue-grey leaf, and touched his ring to spark the roll alight. He drew on the smoke, which drifted up sweet and greyish.

David Langford's "A Surprisingly Common Omission" is a drabble written without using the letter E.
Harlan Ellison once sat in a department store window for five hours, with the challenge being that he write a 500+ page novel in that time-frame. And pulled it off. In addition, he's written several short stories in this manner, with other people providing written prompts just before he starts.
NaNoWriMo's idea is to write a novel with at least X words in a month. The default value of X is 50,000, but some people go for up to million words.
- Similarly, Script Frenzy is a challenge to write 100 pages of script.
- On the other end of the spectrum, there's flash fiction, short stories under a certain length, and "drabbles", stories exactly 100 words long.
Dr. Seuss wrote The Cat in the Hat as a challenge to write an interesting story using a very small vocabulary of words which a 6 year-old should know.
- Speaking of Dr. Seuss, reportedly Green Eggs and Ham was a challenge to create an interesting children's story using no more than 50 different words.
Isaac Asimov wrote Insert Knob A In Hole B live on television. While he admitted he saw the challenge coming and prepared for it, the preparation time was a few minutes before the show started.
Jim Butcher wrote the Codex Alera on a bet that he could write a story using what he thought were two terrible ideas: The Lost Roman Legion and Pokémon.
Poul Anderson wrote the essay "Uncleftish Beholding" in which he described basic atomic theory and the periodic table in a manner as if English had never adopted any French, Latin, or Greek vocabulary but instead only used its Germanic roots:

Some of the higher samesteads are splitly. That is, when a neitherbit strikes the kernel of one—as, for a showdeal, ymirstuff-235—it bursts it into lesser kernels and free neitherbits; the latter can then split more ymirstuff-235. When this happens, weight shifts into work. It is not much of the whole, but nevertheless it is awesome.

Alphabetical Africa is a novel in 52 chapters, beginning with only words that start with 'a', and then 'a' and 'b', up to chapter 26, where all the alphabet can be used. From chapter 27 to 52, the letters words can start with recede back to 'a'.