Fixpoints are one of the most important concepts in denotational semantics and domain theory (see [0] for a good introduction). Denotational semantics lets you precisely specify the meaning of a program as a mathematical object, and also nicely integrates partiality and nontermination. It's quite motivating to learn order theory this way as well.

[0] https://en.wikibooks.org/wiki/Haskell/Denotational_semantics