The Istari essay in Unfinished Tales answers this: But the last-comer was named among the Elves Mithrandir, the Grey Pilgrim, for he dwelt in no place , and gathered to himself neither wealth nor followers, but ever went to and fro in the Westlands from Gondor to Angmar, and from Lindon to Lórien, befriending all folk in times of need.
Also the headnote to the Tale of Years for the Third Age in RotK: Mithrandir was closest in friendship with the Eldar, and wandered mostly in the West, and never made for himself any lasting abode .
In so much as Gandalf could be said to have a home, it is the Gardens of Lórien in Aman, where he lived as Olórin; see the Valaquenta:
Wisest of the Maiar was Olórin.
He too dwelt in Lórien, but his ways took him often to the house of Nienna, and of her he learned pity and patience.
Even then he did journey about somewhat, as the quote given above shows, and also further from Unfinished Tales:
Then Manwë asked, where was Olórin?
And Olórin, who was clad in grey, and having just entered from a journey had seated himself at the edge of the council...
Lórien in Aman, which should not be confused with Lórien in Middle-earth, is presumably where Gandalf eventually returns to at the end of Lord of the Rings.
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It's an interesting question.
His home would definitely be considered Aman, where he returns at the end of the Lord of the Rings.
However I assume you mean within Middle Earth.
We know Saruman had set himself up in Isengard and Radagast has a home called Rhosgobel, so it's already more common for Wizards to have a home than not.
However Gandalf is a special case, and I don't think he had a home beyond residing for a time in places like Rivendell or Lorien.
He's travelling for over a year solid in the course of the Hobbit: March 15, 2941
- Thorin Oakenshield meets with Gandalf the Grey at the Prancing Pony in Bree, the Quest of Erebor begins July, 2941 - Bilbo Baggins obtains the One Ring; the White Council drives Sauron out of Dol Guldur October, 2941
- Esgaroth is attacked by the dragon Smaug, who is consequentially killed by Bard the Bowman; Thorin Oakenshield, Fíli, and Kíli killed at the Battle of Five Armies along with Bolg son of Azog; Dáin II Ironfoot becomes King of The Lonely Mountain; Town of Dale reestablished by Bard June 22, 2942 - Bilbo Baggins returns to Bag End
He'd been hunting Gollum on and off for 17 years by the start of the Lord of the Rings, and then obviously spends close to 18 months travelling in the course of the Lord of the Rings: 3001 - Bilbo Baggins turns 111, passes the One Ring on to Frodo Baggins, and leaves the Shire.
From now on Aragorn and Gandalf intermittently hunt Gollum April 11, 3018 - Gandalf reaches Hobbiton.
He returns to the Shire, telling Frodo Baggins he must take the Ring away September 21, 3019 - Hobbits arrive in Rivendell.
At no point in any of this do we hear mention of a home, nor is there reference to upkeeping of one in his absence or any concern for what might be happening to it.
I'd conclude that Gandalf didn't have a fixed place of residence in Middle Earth.
All dates sourced from the Timeline of Arda .
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One hint that he has no home is that one of his names is "The Grey Pilgrim"
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These first two quotes were provided in another answer, but I include them here to build on their line of thought, because there are a few other relevant quotations that give us a definitive answer to Gandalf's dwelling and why .
From Unfinished Tales :
But the last-comer was named among the Elves Mithrandir, the Grey Pilgrim, for he dwelt in no place , and gathered to himself neither wealth nor followers, but ever went to and fro in the Westlands from Gondor to Angmar, and from Lindon to Lórien, befriending all folk in times of need.
From
The Return of the King : Mithrandir was closest in friendship with the Eldar, and wandered mostly in the West, and never made for himself any lasting abode .
The wording in the second is closely echoed in a passage from The Silmarillion , in the chapter Of the Rings of Power and the Third Age .
First, when describing the arrival of the Istari into Middle-earth: Mithrandir was closest in counsel with Elrond and the Elves.
He wandered far in the North and West and made never in any land any lasting abode … And in the next paragraph: Mithrandir refused the office [of head of the White Council], since would have no ties and no allegiance, save to those who sent him, and he would abide in no place nor be subject to any summons.
Notice the surrounding context of each passage:
Gandalf (i.e. Mithrandir) never settled in a permanent home, was closest of the Wizards with the Elves, accumulated no wealth, did not seek out followers, and refused defined leadership roles.
Based on these passages, Gandalf was explicitly transient, and for a reason: he did not settle down anywhere because he did not want anyone thinking he specifically 'owed' them his service simply for living in their land.
His service was due only to the Valar, so it is reasonable that he considered only Valinor to be his true homeland.
The closest places Gandalf had to a 'home' was when he resided with the Elves of Elrond or Galadriel, because they most of all understood who he was (a Maia) and why he needed to come and go (he was sent by the Valar to guide and help all of Middle-earth against Sauron's evils).
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Gandalf has many times expressed an affection for hobbits; the hobbit children are raised on stories of him.
It is at least mildly implied that when Gandalf is not off seeking knowledge/questing/kicking ass, which is not exactly often, he prefers to hang his hat so to speak in the shire.
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Does Gandalf have his own home or does he just live where he goes?  Is there a place that he prefers to live and calls home most of all?
Where does Gandalf Live?
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<the-lord-of-the-rings><tolkiens-legendarium><gandalf>
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 In so much as Gandalf could be said to have a home, it is the Gardens of Lórien in Aman, where he lived as Olórin. However, there is a hint that he has no home in "The Grey Pilgrim" and in the Istari essay in Unfinished Tales. Gandalf lives wherever he goes; he has no fixed abode.
Gandalf lives wherever he goes; he has no fixed abode.