Has any publisher ever released a complete set of Asimov's short stories? If not, which set comes closest to being complete?
A complete collection of all of Asimov's fiction was planned, but abandoned after only two volumes.
According to Isaac Asimov in the introduction to The Complete Stories, Volume 1, Doubleday intended to publish a uniform collection of all of his fiction: short stories and novels, science fiction and mysteries:
It is time, therefore, for Doubleday to pull the strings together and get all my fiction—short stories and novels, too—into a definitive form and in uniform bindings, both in hard and soft covers. […]
It is Doubleday's intention to make this multivolume collection definitive and uniform in the hope that the science fiction public, the mystery public (for my many mysteries will also be collected), and libraries as well will seize upon them ravenously and clear their book shelves to make room for Isaac Asimov: The Complete Stories.
This means that there would have been many volumes, and not just three, as it is sometimes reported.
Doubleday published The Complete Stories, Volume 1 in 1990 and The Complete Stories, Volume 2 in 1992. Then Asimov died and the series was abandoned, for reasons unknown. Together the two published volumes contain about 80 of Asimov's ca. 400 short stories.
So what can you do? The website The Fiction of Isaac Asimov has a guide what you need to do to read all of Asimov's short stories. It's too long to copy it here, but essentially you have to get 29(!) of his collections/books, and then you are still left with about 40 short stories that were never collected and that you have to track down.
The closest thing I can find is the 2-volume Asimov's Complete Stories published by Doubleday, though with only 80-odd stories it doesn't really look all that "complete."