One I’ve been interested in for a while now is Elaine May’s “Mikey and Nicky”. Paramount took the film away from her after she spent too long in post-production, and released a theatrical cut which she more or less disowned. Over the course of the next decade, one of the producers somehow re-acquired the film from Paramount, and hired an editor to approximate May’s cut as best they could. Although I don’t think May was actually involved in the edit, she did approve the final cut; it appears to have premiered at a film festival in 1986 or so, after which only this cut has ever been seen.
From what it sounds like, the theatrical cut wasn’t as good, but I believe there is alternate footage. I don’t know if there is any release of the original cut; if there is, it would have to be a pretty old VHS or LD (or out-dated format). Wikipedia also adds a new wrinkle – it makes the unsourced claim that the “director’s cut” released by Criterion on blu-ray is different than the 1986 director’s cut. I don’t know if that’s accurate or not, I hadn’t heard anything, but information on this movie (especially from the pre-Internet era) seems very skimpy on-line.