When the very adjacent sentence is “Poets have to write poems”, I think this question is like asking “what kinds of poems, not for the sake of poems, can you actually write…”, then ask about how they change the world, etc.
Even poets don't write poems just for the sake of poems — they write because it gives them pleasure, because they are moved by some strong feeling or because they have some irresistible urge to say something, etc. (And sometimes because they're being paid for it, or hope to be.) Anyway, here are some reasons (in no particular order, and not mutually exclusive—there's some overlap) for Knuth to write programs:
1. Because it gives him pleasure.
2. Because he wants to find out the answer to some question — for instance, how many knight's tours are unchanged under a 180° rotation?
3. Because he wants to experiment or gain more experience with some algorithm(s) or method(s) that he's learning or writing about.
4. Because he wants to collect data on their performance, so that as a scientist when he makes a statement like “a program using this algorithm finds the answer for N=13 using only 0.3 billion memory accesses”, it is a correct statement.
5. Because he wants to understand something better (e.g. his program for linear programming where he says he understood the simplex method clearly only after implementing it).
6. Because he wants to "debug" or interactively see what sub-components of some algorithm do.
7. Because he has encountered (or been posed) a hard problem and wants the challenge of solving it.
8. Because he has seen a beautiful program and wants to translate it to his preferred style, as he'd like more people to appreciate its clever ideas. (E.g. his reimplementation of the original "Colossal Cave Adventure" game of Crowther and Woods.)
9. Because he wants to provide this program as an illustration of some idea or algorithm mentioned in his books.
10. Because someone has asked him for the program.
One can easily imagine more reasons. Though at the rate of five a week it would be more than 7500 programs over the last 30 years, he has given a few examples on his webpage: https://cs.stanford.edu/~knuth/programs.html (Many of them are in CWEB so to help people who don't have `cweave` installed, I typeset them in 2017: https://github.com/shreevatsa/knuth-literate-programs/tree/m... — I ought to update the repository sometime.)