We might identify a number of "great questions" posed by Melville's novel.
Thematically the novel presents questions relating to man's relation to his fellow man in terms of moral responsibility. We see this in Ahab's ethical relationship to his crew (should he press the crew to pursue such a dangerous mission?). We see this also in Quequeg's rescue of his fellow harpooner in the whale's head and in the relationship between Ishmael and Quequeg.
Another thematic question might be categorized as referring to man's relationship to nature and God. The whale is closely identified with nature in the text and Ahab articulates his quest as being one of overturning any natural order that would subject him to the will of another, even if this other is God. This dynamic can be understood as relating to the issue of free will as well.
Formally, the novel asks a number of interesting questions as well. Can a novel be coherent when written in varying styles? Can a novel remain coherent when narrated by multiple voices (which are not coded or idenitified when they switch)?
...often during the middle section of the voyage Ishmael’s voice recedes and the reader is presented with a traditional, omniscient narrator’s view of events, with the consequence that the author, Melville, and the character Ishmael become identified as one and the same in many readers’ minds. Shakespearean soliloquies and learned discourses on whaling history and anatomy are used to break up the narrative thread. (eNotes)
Formally, this novel is quite experimental and uncommon and poses questions about what is formally necessary for a novel to be "a novel".