From Chris Miller's paper in 1995, the structure $(\mathbb R_{\mathrm{an}}, (x\mapsto x^r)_{r\in\mathbb R})$,is the largest known polynomially bounded o-minimal structure as of that time. I wonder if it has been confirmed that it is the largest one or new examples have been found.
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There have been constructions of other polynomially bounded o-minimal expansions of the real ordered field. The first main example is by van den Dries and Speissegger in 1998 (The real field with convergent generalised power series), and the lastest one is by Rolin, Servi and Speissegger (Multisummability for generalized power series).
They both only add functions defined on compact subsets of $\mathbb{R}^n$, so they do not include the total power functions (they might still be definable there). What the first one adds is functions given by convergent series with real powers, whereas the second one adds functions which have a possibly divergent asymptotic expansion, i.e. are sums of "multisummable" series.
I'm not sure it's easy to show that adding them preserves polynomial boundedness, but I could get back to you on that (as I suspect it should not be too difficult).
