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Integral exponential functions are the members of the least class of real functions containing      1 1      , the identity function, and closed under addition, multiplication, and binary exponentiation sending      f f       and      g g       to      f g {f^g}      . This class is known to be wellordered by the relation of eventual dominance. It is shown that for each natural number      n n       the order type of the integral exponential functions below      2 x n {2^{{x^n}}}       (below      x x n {x^{{x^n}}}      ) is exactly      ω  ω  2 n −  1 {\omega ^{{\omega ^{2n - 1}}}}       (     ω  ω  2 n {\omega ^{{\omega ^{2n}}}}       respectively). The proof, using iterated asymptotic expansions, contains also a new proof that integral exponential functions below      2 2 x {2^{{2^x}}}       are wellordered.

Fine structure of the integral exponential functions below 2^{2^{𝑥}}