Where does Gandalf live

[Legate of Amon Lanc]

It is pretty clear that Gandalf does not have any "permanent residence", and I think in the essay about the Istari in Unfinished Tales it is maybe even stated explicitely (but even if not, clearly enough). I think it is at least implied also that the Wizards' "settling down" was partially also a failure, both in case of Saruman (who wanted to become a "politically established power" and also wanted to "show off" with his "Wizard's tower") and Radagast (who just chose a shelter to hide in among his favourite animals). A Wizard was supposed to move to be able to assist in different places, if I overdo it a little bit, I would liken them to wandering apostles - they were supposed to boost the morale wherever they came, and simply relied on the fact that the locals would offer them a shelter for the time being. (That's also the reason why Gandalf evidently stopped going too far East and South, since he didn't receive the welcome there.)

That is not to say that Gandalf clearly had a few favourite "retreats" where he was certain of being received - such as Rivendell, Lórien, or also the Shire (in a different way) where he could go "on a holiday" to please himself with looking at happy hobbits who had no idea about any shadow and evil. But obviously, his task always drove him on, so he did not have time to settle down permanently anywhere.