I might suppose that many novels began as published short stories when their authors thought “Hey, I can see how this could be added to or expanded,” but for the life of me, I can’t think of many examples, only “Flowers for Algernon.” Can you?
Ray Bradbury’s Fahrenheit 451 started out as 2 different short stories that became a novella and finally the novel.
Greg Bear expanded his 1983 short story “Blood Music” into a novel.
Joanna Russ’s “When It Changed” was expanded to become The Female Man
Arthur C. Clarke wrote a novella called Against the Fall of Night (1948) which he revised and expanded into a novel with the same title (1953), then revising and expanding it again with the title The City and the Stars (1956).
Arthur C. Clark’s The Sentinel was reworked into 2001: A Space Odyssey.
John Varley’s Millennium (see also the K. Kristofferson/C. Ladd movie) was based on his own short story “Air Raid.”
The OP didn’t rule this out, so how about we go back aways? E. E. Smith’s “Lensman” series was retrofitted with all of the stories in Triplanetary by the time they saw print as novels.
Orson Scott Card’s Ender’s Game was first a short story.
The Voyage of the Space Beagle, The Martian Chronicles, The Dying Earth, and a number of Asimov’s novels were independent stories that were combined together into a “fix-up” (a term coined by A. E. van Vogt) of separate short stories or novella, which was actually pretty common with early sci-fi and fantasy writers when book publishers were reluctant to publish anthologies. Note that this different from the serialized publication of novels that was common with science fiction and fantasy in the “Golden Age”, although that was also related to the reluctance of book publishers to promote sci-fi and fantasy work.
Stranger
Larry Niven fixed up his short story “Rammer” into the first chapter of his novel A World Out of Time.
Interesting that all the examples given in this thread so far have been science fiction.
Asimov’s Nightfall and The Bicentennial Man were also later expanded into novels, though I think in both cases most of the work was done by another writer.
Several of Philip K. Dick’s stories started off as short stories, then got turned into movies, and then had novelizations written of the movies (usually by different authors).
Here are 3 stories that are definitely not science fiction.
Thomas Mann’s Felix Krull: Confessions of Felix Krull - Wikipedia
And F. Scott Fitzgerald described his short story “Winter Dreams” as “a sort of first draft of the Gatsby idea.” It was written in 1922, 3 years before Gatsby. Winter Dreams - Wikipedia
I can’t find a source for it, but Mary O’Hara’s My Friend Flicka, a 1941 children’s novel which inspired several film and TV versions, was adapted and enlarged from a short story.
Carrie by Stephen King was a short story which he threw away – feeling he knew nothing about high school girls and having periods. His wife, Tabitha, saved the day (and quite possibly his career) by fishing it out of the trash and telling him to have another try.
Most examples are from science fiction because they used to do it all the damn time. Get money from the magazine for selling a short story and then get more money by expanding it or appending to it for the novel. There must be thousands of examples. Roger Zelazny did it regularly as a younger writer. “…And Call Me Conrad” was expanded into novel form as This Immortal. “He Who Shapes” was expanded into The Dream Master. Two novelettes, “Dawn” and “Death and the Executioner”, turned up as chapters of Lord of Light. The short version of Damnation Alley was expanded into a novel of the same name.
Mystery writers have a tougher time doing this: the murderer is the murderer. Some exceptions came back in pulp days, when the action is pure adventure rather than whodunit, and so could be expanded. This happened on occasion in the early days of paperbacks, which overlapped with and then caused the death of the pulps.
Mainstream writers often do take a short story and expand it into a novel. Usually the short story is published in a literary magazine. E.g., in 1923 Virginia Wolff published “Mrs. Dalloway in Bond Street” in a small literary magazine called The Dial. Two years later she expanded it into Mrs. Dalloway. Larger magazines got into play as well. J. D. Salinger published “I’m Crazy” in Colliers in 1945 and incorporated the story into Catcher in the Rye.
Tell the truth: if I’d used that word, you’d have rapped my knuckles with a ruler. 
Science fiction also has a history of “fix-ups” where an author turned several short stories into a novel. The first two books of Asimov’s Foundation novels were that. So was James Blish’s Cities in Flight.
Clarke’s Childhood’s End was expanded from a short story.
From literary works, Vance Bourjaily wrote The Man Who Knew Kennedy with one section a short story published before JFK was assassinated.
The Dan Simmons short story Remembering Siri (published 1983) grew into Hyperion (published 1989) and was integrated as part of a chapter in the novel.
No, no, no, never!
Both the Hugo and Nebula Awards have a “Best Novelette” category, spelled that way. I’ve lived all my reading life with Novelette as the spelling of the word.
I know some people spell it novelet, but that’s not the sf way. Or my spellchecker’s, it seems.