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Given some $f(x)$ composed with itself $n$ times, how would one go about finding a closed-form expression in terms of $x$ and $n?$

Specifically, I'm trying to find a function in two natural number variables, $f(x,y) = x^{(x^{(...^{(x^x)})^...)})} =$ "$x$ tetrated to the $y$".

Any resources for learning more aabout function composition at an undergraduate level would be very much appreciated

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    I have an old, very basic, but also amateurish article which I should rework one day... But for a beginning and to get an initial idea and some working examples for a start, you could look into http://go.helms-net.de/math/tetdocs/ContinuousfunctionalIteration.pdf . I did the examples with the software Pari/GP which can handle matrices, rational numbers and power series. With something like that you should be able to re-calculate the examples (if needed at all). – 2017-02-04

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This doesn't have a closed form solution unfortunately.

Wikipedia has some good information on function iteration, and has a section on techniques to find closed forms.

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    This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - [From Review](/review/low-quality-posts/756446) – 2017-02-03
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    @HarshKumar The question asks for a closed form solution. There isn't a known closed form solution. – 2017-02-03