My impression is that computing (the homotopy groups of) $\operatorname{Diff}(n)$, the diffeomorphisms of the n-disk, is a hard problem.
For example, the connected components are related to exotic spheres. However, I think $\operatorname{Diff}(n)$ can be stabilized: there is a map of spaces $\operatorname{Diff}(n) \rightarrow \operatorname{Diff}(n+1)$ that sends $\phi$ to $\phi \times Id_{[0,1]}$ where we make a fixed identification $D^{n+1} \cong D^n \times [0,1]$. Let $\operatorname{Diff}^{\textrm{stable}}$ denote the colimit $\lim_{\rightarrow} \operatorname{Diff}(n)$.
Question What is known about $\operatorname{Diff}^{\textrm{stable}}$? Can its homotopy type (or homotopy groups) be computed in terms of something simpler?
Answers in the form of references would also be appreciated!