Where did Gandalf live?

[Alcuin]

Return of the King, “Appendix B”, just before the Tale of Years of the Third Age, there is a passage discussing the Istari which states that

Mithrandir was closest in friendship with the Eldar, and wandered mostly in the West, and never made for himself any lasting abode.

This passage is cited in Unfinished Tales in the essay “The Istari”, which says that

...the last-comer [of the Istari] was named among the Elves Mithrandir, the Grey Pilgrim, for he dwelt in no place, ... but ever went to and fro in the Westlands ...., befriending all folk in times of need. ... Mostly he journeyed unwearingly on foot, leaning on a staff...

Gandalf was in particular alliance with Elrond in Rivendell and Galadriel in Lórien: when he was returned to Middle-earth after his battle with the Balrog of Moria, he had Gwaihir the Windlord take him to Lórien. But the two texts (in Return of the King and Unfinished Tales) would seem to indicate that these were places of rest and recuperation, much as was the Shire in the passage cited by A_Brandybuck.