Since $\K$ is not closed under exponentiation, it is fair to ask whether the Liouville closure of $\K$ is again an Ilyashenko field. This appears to be the case; indeed, working with Margaret Thomas, we conjecture that if $(\la, \M, T)$ is any Ilyashenko field with $\M$ the set of (not necessarily convergent) LE-monomials, then the Liouville closure of $\la$ is again an Ilyashenko field with respect to the set of LE-monomials $\M$.
The thornier question is that of the Newtonian closure in the sense of Aschenbrenner et al.; I do not yet know how to deal with this.