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Existence of an o-minimal structure defining a transexponential function

Establish whether there exists an o-minimal structure on the real field that defines a transexponential function E: R -> R such that for every n ≥ 1, E(t) eventually dominates the n-fold iterate of the exponential, i.e., E(t) ≥ exp^n(t) for all sufficiently large t.

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Background

Known o-minimal structures are exponentially bounded, meaning they do not define functions that grow faster than all finite iterates of exp. Determining whether transexponential growth can be captured within o-minimality would clarify the upper limits of growth behavior compatible with tame geometry.

Progress on related Hardy field questions exists, but a corresponding o-minimal structure achieving such growth has not been constructed or ruled out.

References

It is unknown if one can achieve growth rates faster than every iterated exponential in an o-minimal structure, i.e., every known o-minimal structure is exponentially bounded: Open Question 3.31. Does there exist an o-minimal structure which defines a transexponential function? I.e., a function E : R -> R such that for every n ≥ 1, eventually E(t) ≥ expn(t) as t -> +00, where exp1 := exp and expn+1 == exp o expn.

— Deep Learning as the Disciplined Construction of Tame Objects (2509.18025 - Bareilles et al., 22 Sep 2025) in Open Question 3.31, Section 3.3 (Tame asymptotics)