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In "The Two Towers", in the chapter "The White Rider", Gandalf describes his battle with the Balrog. He mentions being underground, deeper than even where the dwarves went.
'We fought far under the living earth, where time is not counted. Ever he clutched me, and ever I hewed him, till at last he fled into dark tunnels. They were not made by Durin's folk, Gimli son of Glóin. Far, far below the deepest delvings of the Dwarves, the world is gnawed at by nameless things. Even Sauron knows them not. They are older than he. Now I have walked there, but I will bring no report to darken the light of day. In that despair my enemy was my only hope, and I pursued him, clutching at his heel. Thus he brought me at last to the secret ways of Khazad-dûm: too well he knew them all. Ever up now we went, until we came to the Endless Stair.'
Clearly Gandalf does not want to describe this place any further (he describes this place as a "despair", which implies it's not something he'd readily mention again), so I doubt he brings it up again later in the book, or in Return of the King (although I haven't read past "The White Rider" yet, and I certainly wouldn't remember something this obscure from the last time I read these books years ago).
Is this place mentioned anywhere else in any of Tolkien's other works, such as the Silmarillion? Any and all further information on this place would be appreciated, as well as the "nameless things" that Gandalf mentions.
The very similar question What is the place that Gandalf falls to after the battle with the Balrog? identifies that it is some kind of lake, but doesn't add any further information on it besides what's in the text I've quoted. Specifically, I'm looking for any further information outside of this chapter.
I've just found this question about the creatures, which I wanted to update this question with, since they are related.