Kuiper's theorem is well-known to give the triviality of the homotopy groups of ${\rm GL}(\mathcal{H})$ for $\mathcal{H}$ a (separable) infinite-dimensional complex Hilbert space. Work of Palais later gives sufficient control over the infinite-dimensional manifold structure so that we know ${\rm GL}(\mathcal{H})$ is in fact contractible. Mathai Varghese asked me about ten years ago (as a student) if I would be interested in proving that ${\rm GL}(\mathcal{H})$ was holomorphically contractible. This was not a challenge I took up at the time, though it was suggested to me that a first step would be to think about whether $\mathcal{H}\setminus\{0\}$ was holomorphically contractible.
Is this something reasonable to expect? The first issue is to figure out in what topology we even have a complex infinite-dimensional manifold, if any.
The reason that it would be useful to know is that ${\rm GL}(\mathcal{H})/\mathbb{C}^\times$ would be a holomorphic $K(\mathbb{Z},2)$. I've always wanted to know if this is possible...