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In the field of Denotational semantics they have developed a (very) formal definition of iteration in terms of recursive functions that you might like to look into. https://en.wikipedia.org/wiki/Denotational_semantics


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The $k$th iteration is just the $k$th element of some appropriate sequence. In general, I doubt it's a formally defined concept; it's just one of those things whose meaning is basically the natural-language meaning of the word.


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