Asking to visit a university, and be given workspace for a week
I disagree with the existing answers. I don't see this as a "big ask" at all, as long as your expectations are reasonable. You should not expect to get a nice individual office, but if I know you at all I would definitely be able to find you a space somewhere in my or my division's labs. My students have also done similar visits in the past, sometimes in labs that we did not have extensive connections to, so I guess other people also don't see this as a huge deal.

Of course you should have some idea what you and the host will get out of the visit - people may not be thrilled to host you if this is really just a cover-up for a private visit. Conversely, for a week-long visit, you should also not expect to be "entertained" for the entire week. People will not be able to cancel everything for an entire week.