The first version of my preprint on quasianalytic Ilyashenko algebras is available on arXiv. Feel free to leave comments here!
Here is the abstract: I construct a quasianalytic field $\F$ of germs at $+\infty$ of real functions with logarithmic generalized power series as asymptotic expansions, such that $\F$ is closed under differentiation and log-composition; in particular, $\F$ is a Hardy field. Moreover, the field $\F \circ (-\log)$ of germs at $0^+$ contains all transition maps of hyperbolic saddles of planar real analytic vector fields.
A more detailed description of its contents can be found in this post.