This is such a big topic that there’s a Wikipedia page devoted to it. I note, however, that it is by no means complete. I’m going to restrict myself to ones I’m familiar with
Before I begin, I’m trying to keep this to novels that the original author came close to finishing, and actually wrote the beginning to, but he/she died or something, and never finished it, so someone else did. I’m not talking about cases where someone else wrote a novel using only the original author’s notes or outlines (The way, for instance, Spider Robinson used Robert Heinlein’s outline and notes to produce Variable Star, no part of which Heinlein himself wrote). Of course, sometimes things are ambiguous, or it’s hard to tell. And I expect that many entries will tend that way if this thread goes on for any time, due to “mission creep”.
The Mystery of Edwin Drood by Charles Dickens – probably the “poster child” for this idea. Dickens not only left it unfinished at his death, he didn’t tell us what the solution was going to be. Needless to say, lots of people have furnished endings, including four movie versions and at least one play.
Almuric by Robert E. Howard – I didn’t know for a long time that Howard never finished his John Carter-esque “planetary romance”, but he apparently left it unfinished. In fact, some people suspect he never wrote any of it, and that it was produced Variable Star-like, from his notes. That seems unlikely, for lots of reasons. But people are on firmer ground speculating about who it was that finished it. Candidates include Otis Adelbert Kline (Howard’s agent, and an author of Carsonesque novels himself), Farnsworth Wright (editor of Weird Tales, where it was published) and Otto Binder (author of the Adam Link stories, among others).
Micro – Michael Crichton left this unfinished at his death, and Douglas Preston (half of the team responsible for Relic and the other Agent Pendergast stories, not to mention several series of fantastic adventure novels written solo) finished it up. I started reading it, but gave up when I could see exactly where it was going.
The Mysterious Stranger – Mark Twain wrote at least three different versions of this story, never quite finishing it. After his death Alfred Bigelow Paine, who wrote Twain’s biography, cobbled together parts of two different versions to produce the edition published shortly after his death. Paine had similarly juggled various autobiographical writings of Twain’s to produce the published “Mark Twain’s Autobiography” that was not really what Twain had intended. In more recent years the original texts of The Mysterious Stranger have been published, and the full and correct Autobiography of Mark Twain has beemn published in three massive volumes starting in 2010, the centennial of his death (and the date Twain wanted it published)
Hornblower During the Crisis by C.S. Forester. This was the last Hornblower novel, which Forester was working on when he died. It was published incomplete in both hardcover and paperback , along with two short Hornblower stories. But sinxce that time there have been several different versions of the ending. Amother ending was suggested (but not written up as a novel) in C. Northcote Parkinson’s The Life and Times of Horatio Hornblower
The Last Tycoon by F. Scott Fitzgerald – I was unaware that this famous Fitzgerald novel hadn’t been finished. Nobody seems to have finished it, either. I haven’t read it, myself
Billy Budd by Herman Melville – I wasn’t aware that this famous title was left unfinished, either. I’ve read some of it. Apparently published in its incomplete state.
The Assassination Bureau, Ltd. by Jack London. I know this mainly from the 1969 film starring Oliver Reed and Diana Rigg (not to mention Telly Savalas and Curt Jurgens – The cast makes it feel like Steampunk James Bond). London bought the story idea from Upton Sinclair (!), wrote 20,000 words (influenced by G.K. Chesterton’s The Man Who Was Thursday), then couldn’t figure out how to end it. Crime writer Robert L. Fish (who wrote the story Bullitt was based on) wrote an ending to it in 1963, long after London’s death.
The Journal of Julius Rodman and The Light-House – Both started by Edgar Allan Poe. Not finished by him, or anyone else, although they’ve been published. I hadn’t even known about these. Now I have to look them up. Poe did write one complete novel, which I’ve read – The Narrative of Arthur Gordon Pym of Nantucket, but it ends in such an inconclusive way that it feels unfinished. Intriguingly, the ideas were continued by two other authors of weird fiction – Jules Verne and H. P. Lovecraft.