I was wondering if Eru has a palace, because the Valar sort of did.
And in case he does, in what kind of architectural style (Elven, Númenórean, etc...) is it made?
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While it's correct to say that Eru doesn't "exist" in the physical sense, there is evidence in The Silmarillion "Ainulindalë" that he does have some form of residence, yes:
[...] and the places of the dwelling of Ilúvatar were filled to overflowing [...]
He seems to have a throne (Tolkien doesn't say but I assume a throne):
But now Ilúvatar sat and hearkened, and for a great while it seemed good to him [...]
And there is also mention of dwellings he made for the Ainur:
But Ilúvatar arose in splendour, and he went forth from the fair regions that he had made for the Ainur [...]
However the nature of these dwellings is not further expanded on, and we probably shouldn't think of them in terms of physical architecture. That's not to say that they're not physical; they certainly could be, but they could just as easily be made of light or some other ethereal substance.
Since Eru and these dwellings existed before the world, even if physical, they certainly would not be in any architectural style that only came to being within the world.
Eru (AKA Ilúvatar) didn't have a palace (at least not in the physical sense), because he was a spiritual being:
Ilúvatar is the Lord for Always who dwells beyond the world; who made it and is not of it nor in it, but loves it.
The Book of Lost Tales, Part One, "The Music of the Ainur"
The Valar, on the other side, had it because they were Eru's emissaries on Arda, sent by him to shape the world:
[T]he fourteen Ainur who witnessed the Vision of Ilúvatar and came to shape Arda were the Valar. By chosing to enter Eä after its creation, rather than remaining in Ilúvatar's Timeless Halls, the Valar agreed to be bound to it until the end.