Final answer:
Francis Ford Coppola was close to being fired from his role as the director of The Godfather several times, with at least five documented instances due to disagreements over casting, production issues, and budget concerns, despite the subsequent success of the film.
Step-by-step explanation:
The question asks about the number of times director Francis Ford Coppola was on the verge of being fired during the production of The Godfather. Coppola faced numerous challenges and conflicts with Paramount Pictures, and it is reported that he was at risk of being fired several times. Specifically, there were at least five documented instances wherein Coppola's job as director was in serious jeopardy. This was due to disagreements over casting decisions, production delays, budget concerns, and the studio's lack of confidence in Coppola's vision.
Coppola's determination to cast Marlon Brando as Don Vito Corleone and Al Pacino as Michael Corleone were major points of contention with the studio. Moreover, his ambitious cinematic style, which looked unconventional to the executives, also put him at risk. The eventual success of The Godfather, which became a cinematic masterpiece and an iconic film in American culture, vindicated Coppola's artistic choices and solidified his reputation as one of Hollywood's great directors.