There’s Michael Moorcock’s Dancers at the End of Time series; more whimsical Moorcockian fantasy than hard sf, though.
That’s Poul, as in Poul Anderson, already mentioned.
Yes, Manifold: Time does go through several universes. Ring doesn’t go quite to the end, but goes quite a bit into the future.
In Slaughterhouse Five we learn that the Tralfmadorians cause the universe to end when some space pilot pushes the wrong button.
The trouble is that even a billion years out is nowhere close to the end of the universe. If that far counts, we have The City and the Stars by Clarke, set a billion years from now, plus Twilight by Campbell, also set in the far future. But not as far as the end by any means.
While Blish is known for the ST books, he was much more famous in SF for the Cities In Flight books and A Case of Conscience, for which he won a Hugo. Both well worth reading, although the Cities books are the only ones that fit this thread.
Jack Vance’s Dying Earth books are set on Earth as the sun is flickering and dying; whether you consider them SF is another thing, as they are essentially magical fantasy. Great reads though.
The Doctor Who story *Utopia *takes place in the dying days of the Universe.
Book four in the original series of Neil Gaiman’s The Books Of Magic The young student magician Timothy Potter travels to the far future and sees the final death of the universe.
Between the Strokes of Night by Charles Sheffield has humanity using a combination of cryogenics and a special ultra-slow metabolic rate and consciousness at ultra-cold temperatures to approach the Big Crunch.
Job: A Comedy of Justice by Robert A. Heinlein is an SF novel that postulates the Bible as literally true, and follows a protagonist through the Trump and the Shout and God ending His Creation.
*** Ponder
Job is actually a pastiche of James Branch Cabell’s works, especially Jurgen, A Comedy of Justice. Although it does fit into the thread here.
What I like about SDMB: as soon as I get ready to add my 2 cents’ worth, someone else phrases it better than I ever could.
I didn’t think man would still be around at the end of the universe, though insofar as I’m concerned the universe ends when I die. Solipsist much? Yeah, when it comes to topics like these.
I last read this story over 40 years ago and don’t remember the title, but there was an issue of Edmond Hamilton’s Captain Future where he and his band of Future Men (the brain, the robot, and the android) travel to the end of the universe to save humanity from some evil mutants. In this tale, the universe is expected to start contracting after it has reached maximum expansion.
And not to spoil the ending in case you plan to read it,
Captain Future destroys the mutants by re-inventing the carbon arc light and mowing them all down with U-V radiation 
.
Hey, when I was 9, I thought it was great!
Curse you! shakes fist
(That was an awesomely eerie sequence. Why isn’t there more Tim Hunter out there? sigh)
Lightray, thanks! I’m gonna ask for that one at Star Clipper; it might be the one I heard about. Really vague, but I understood it was that Alan Moore, and it had all the immortal critters like The Watcher at universal heat death.
???I dunno
If you mean the Books of Magic, it was Gaiman and didn’t include The Watcher, since it was set in the DC universe. It did have John Constantine, the Phantom Stranger, Mister E (who was the one that took Tim to the end of the Universe), and Dr. Occult, along with some guest appearances by various other DC magic types. Really good book.
This is a bit of a spoiler, but it can’t be avoided - just naming the book here is going to do that, so a spoiler box seems kind of superfluous, but I’ll give it a go:
Darwinia By Robert Charles Wilson:Everything in the novel actually takes place in a simulation at the Omega point
and there’s a good term to google for one kind of end-of-universe stories: Omega Point
“The known universe” isn’t just places we have been, but everything we have seen. The death of our sun isn’t going to affect a quasar billions of lightyears away.
Nightland by William Hope Hodgson is set towards the end of the Universe…
Cosm by Gregory Benford features the end of a universe, iirc…
Not well known (it was self published), but parts of The Metamorphosis of Prime Intellect arguably takes place at the end of the Universe. Warning, there are some pretty violent scenes in that book.
Yeah. It’s been a long time since I’ve read it and since re-reading the story
I find it as brilliant as I remembered it. 
“Asimov Rules”
Campbell wrote another story, called “Night”, set untold billions, maybe trillions, of years in the future, with only a few stars dimly glowing at the end of the Universe. Probably not accurate even by 1935 cosmology, but very evocative.
Going through my old books just now, I found Dark is the Sun by Philip Jose Farmer. It’s set 15 billion years from now, and the universe is dying, if not dead.