Absalom, Absalom! by Faulkner
Molloy by Beckett
Jealousy by Robbe-Grillet
The Sot-Weed Factor by John Barth
Replay by Ken Grimwood
The Carpet Makers by Andreas Eschbach
Incredibly difficult question. If pushed, right now, I’d have to say:
A Suitable Boy - Vikram Seth
Midnight’s Children - Salman Rushdie
The Great Gatsby - F. Scott Fitzgerald
The last one may be unnaturally favored because I just finished reading it again last night. It’s a toss-up between that, Freedom by Jonathan Frantzen and A Prayer for Owen Meany. But it is a truly incredible book.
One I read recently that will be a contender in future. And I hope will gain its rightful place as a modern classic. It has the sparse beauty of Hemingway, the psychological insight of Patricia Highsmith, yet in the context of a page-turning, driving plot:
The Outcast - Sadie Jones
If you like literary fiction, get this devastating book, with my wholehearted recommendation.
Cheater!! 
The Crimson Petal and the White by Michel Faber
The Robber Bride by Margaret Atwood
At Swim-Two-Birds by Flann O’Brien
Persuasion, Jane Austen
A Soldier of the Great War, Mark Helprin
Bleak House, Charles Dickens
“Busman’s Honeymoon” - Dorothy Sayers
“When Genius Failed” - Roger Lowenstein
“Into the Wild” - Jon Krakauer
We aren’t doing fiction only, are we? I’m surprised at the lack of non-fiction here. Usually in these types of threads it’s a mix of mostly fantasy and non-fiction.
Needful Things, by Stephen King
From the Corner of His Eye, by Dean Koontz
and the third would have to be a tie between One Door Away From Heaven, by Dean Koontz or I Never Promised You A Rose Garden, by Hannah Green/Joanne Greenberg.
What!? I chose my three. I just happened to explain my thinking. 
The Dollmaker by Harriette Simpson Arnow
The English Passengers by Matthew Kneale
Cloud Atlas by David Mitchell
If the OP had asked for just one, it’d be The Dollmaker, hands down. It’s been almost 40 years since I first read it, and nothing else has even come close to the effect this book has on me.
I think you’re the first person I’ve ever heard say that’s one of Heinlein’s best, let alone one of the best books of any kind. I think I might have read it when I was a teenager, but I remember next to nothing about it. Have to look at it again.
Lipstick Traces: A Secret History of the 20th Century - Greil Marcus
Pop. 1280 - Jim Thompson
We Wish To Inform You That Tomorrow We Will Be Killed With Our Families: Stories from Rwanda - Philip Gourevitch
Little Women by Louisa May Alcott
A Beautiful Mind by Sylvia Nasar
Love, Loss and What I Wore by Ilene Beckerman
My favorite books about childhood, science and crafting a life as a grown woman.
*Lord of the Rings *- The Good Professor himself.
The Moon is a Harsh Mistress - Robert A. Heinlein
The Hobbit - by J. R. R. Tolkien
Cripes. That’s hard. Just fiction, or anything?
Fiction:
Gilead- Marilynne Robinson
The Dispossessed- Ursula K. Le Guin
You know, for the third, I can’t decide between about twenty books. I’ll throw in a non-fiction book: Pilgrim at Tinker Creek. There.
Catch-22 (Heller)
One Flew Over The Cuckoo’s Nest (Kesey)
Desert Solitaire (Abbey)
Double Star may not be RAH’s best, but the OP asked for favorites! I love so many of his books, but something about his making a politician a HERO makes it different.
As best I can tell, Heinlein is the only author listed in this thread with more than one book. I’ve been compiling a list of titles, and Lolita, The Stand, The Lord of the Rings, and The Moon is a Harsh Mistress, are the only books to be mentioned more than once. That really surprises me, I thought maybe there would be more doubling up.
Actually I picked LotR & the Hobbit. So Tolkien got 2 also.
There’s no way I could come up with only three. I am so glad to see I am not the only lover of The Stand and A Prayer for Owen Meany. Both those stand up to repeated re-reading, I’ve found.
Um…
The Stand, Stephen King
I Know This Much Is True, Wally Lamb
Heartbreak Hotel, Anne Rivers Siddons
And thanks, folks–I will be bookmarking this thread in dreams of coming summer, and endless hours between “after work” and “too dark to read” for my blissful time, sitting in the back yard, reading.
Reading.
And reading.
…hey, when the thread dies down, someone compile a list, eh? And post it? Baker?
Ah. Le Guin’s best work. Fourth on my list.