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I wonder if there's a word for these kinds of nested or multi-layer acronyms. Acronyms in acronyms.

It's like a recursive acronym with a base case, so it terminates.

Somebody needs to make a recursive acronym in the shape of the Ackermann function.

It's not recursive, though -- that would only be true if AOL stood for AIM On-Line.

Maybe "nested acronym" is a better term?

> Multi-layered acronyms: AIM: "AOL Instant Messenger," in which "AOL" originally stood for "America Online"