If you took away the crazy stuff — the paranoia, the paralyzing fear of germs, the urine in the milk bottles — would the story of Howard Hughes be worth telling? Or, more pointedly, worth turning into a $100 million event film helmed by one of the world’s greatest living filmmakers and showcasing arguably its most bankable young star?

Hard to say. America has seen a lot of great industrialists come and go, although few who roamed the bedroom with the likes of Jean Harlow, Ava Gardner and Katharine Hepburn, a legendary hat trick by any standard.

What isn’t hard to say is that Martin Scorsese and Leonardo DiCaprio turn Hughes into one of the decade’s most compelling screen heroes in “The Aviator,” a towering, vigorous drama with ambition and resources to match the legendary mogul himself. Certainly, there have been other Hughes projects — a 1977 TV movie with Tommy Lee Jones; “Melvin and Howard,” with Jason Robards playing the billionaire in his demented twilight — but none that so skillfully reconciled his mania with his genius.

After a brief prologue that foreshadows Hughes’ mental difficulties, Scorsese and screenwriter John Logan (“Gladiator”) whisk us to 1928, where the wealthy, 23-year-old tool fortune heir is directing his aerial dogfight epic “Hell’s Angels” — the most lavish and expensive movie production of its time. Unable to get the shot he wants with the airplanes available, this squawky showbiz upstart orders his chief engineer to modify an existing plane, commissioning a new company, Hughes Aircraft, to bury the costs. That, fellow Americans, is the living definition of a self-actualizing capitalist stud: He builds industries by accident.

Later, in a fit of daring, he buys a small domestic carrier called TWA. Between two spectacular aerial sequences — one involving chaotic biplane maneuvers for “Hell’s Angels,” the other, a test-flight fiasco that would leave Hughes burnt and near death — Scorsese snoops around Hughes’ relationship with Hepburn, played with mahhhvelous patrician flair by Cate Blanchett.

Confessing over a round of golf that she “takes seven showers a day,” Hepburn is strangely, compulsively drawn to Hughes, who later tells her that he has “crazy ideas of things that may not really be there.”

Blanchett (whose performance recently garnered a Golden Globe nomination) and DiCaprio have such a freaky, tragic bond, it’s something of a shame that the whole movie wasn’t constructed around the relationship. Harlow (pop starlet Gwen Stefani, a bit awkward) and Gardner (“Underworld’s” Kate Beckinsale) come off as little more than trophies. Scorsese — reteaming with DiCaprio after the wild, burlesque butchery of “Gangs of New York” — is careful to drop hints of illness during even the character’s most seemingly sane moments. Whether it’s his design for Jane Russell’s brassiere or the construction of a massive military transport plane (later dubbed the Spruce Goose), his obsessive drive for perfection forever threatens to boil over, as it does in one chilling scene when sanity slips from his grasp, causing him to repeat the same phrase (“Get me all the blueprints”) over and over.

Shirt stains are a problem, too — DiCaprio spends a fair amount of screen time huddled over restroom sinks, scrubbing feverishly. One suspects that Scorsese might have jumped the gun a bit on Hughes’ foundering sanity, for dramatic effect; after locking himself in a screening room for weeks with nothing but the aforementioned milk bottles for company, it’s no small miracle when he cleans up and goes to Washington to defend himself against a Senate witch hunt engineered by his rival, Pan Am monopolist Juan Trippe (Alec Baldwin). Scorsese is aiming for our indignation as free-market patriots here, and he succeeds — it’s a triumphant sequence, with Alan Alda excellently loathsome as Trippe’s congressional lap dog.

Physically, DiCaprio is less than a perfect fit to play Hughes (at a boyish 30, he looks too callow to play a man in his 40s), but the actor musters enough maturity to get the job done, and done well. “The Aviator” could well mark a turning point for DiCaprio, whose “Titanic” notoriety has somehow eclipsed that fact that he is, by any measure, a remarkably good and focused actor. For Scorsese, “The Aviator” does not represent his most profound or most original work (one detects whiffs of “Ed Wood” and “A Beautiful Mind” throughout), but it could be his most romantic, sweeping and ambitious. And like the Goose, it surely takes flight.