We examine the possibilities for limit functions on bounded Fatou components arising from the non-autonomous iteration of polynomials where one considers compositions chosen from sequences of polynomials with suitably bounded degrees and coefficients. In contrast to the restrictiveness of the classical context where one iterates a single polynomial or rational function, we show that a far wider range of limit functions can be obtained. More precisely, we show that it is possible to obtain the \emph{whole} of the classical Schlicht family of normalized univalent functions on the unit disc as limit functions on a \emph{single} Fatou component for a \emph{single} bounded sequence of quadratic polynomials.
The main ideas behind this are quasiconformal surgery and the feature of dynamics on Siegel discs where suitable high iterates of a single polynomial with a Siegel disc U approximate the identity closely on compact subsets of U. This allows us both to approximate many functions from the Schlicht family on a Fatou component and to correct the small but inevitable errors arising from these approximations. In our presentation, we hope to go more into the details of just how we did this than is normally possible in a shorter talk and to give an idea of the somewhat involved constructions required to obtain this result.