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They're used in a distributed hash table implementation called chord (https://en.wikipedia.org/wiki/Chord_(peer-to-peer)

They're excellent general purpose structures for data sharing of immutable data (e.g., a slightly-changed copy shares much of the same data as the original). Clojure uses them extensively, since immutable data has the nice property of being thread-safe by default.

You can create a persistent ordered container that has both prepend and append in amortised constant time; and concatenate in log time.

Caching isn't great though.

Pagetables are a trie, so it's max log log log time to prepend, append, or concatenate, and usually constant small for two of them (but admittedly: you usually have to pick if you don't know how many arrays you want).

There are several use-case examples in the Readme for this Clojure finger tree implementation:

https://github.com/clojure/data.finger-tree

Just want to add that the data structure behind all of Clojure's default data types are finger trees.

They can be used to implement an immutable deque.