Stephen starts from the very beginning of the series; in Master and Commander, he conducts a trepanation on a crewman who has sustained a dangerous skull fracture. Think for a moment about how risky such an operation was in the days before reliable anesthesia and antisepsis, and aboard a swaying, rocking sailing ship to boot. Now consider that the patient not only survived the operation with flying colors, but returned to full duty. This operation made Stephen's reputation among the crewmen of the Sophie and they clubbed together to buy him a gold-headed cane (one of the symbols of a physician in that period) in appreciation. And then he does it again in The Far Side of the World.
Any time a smaller ship captures one that significantly outguns it is an in-universe CMoA. Case in point, the Sophie vs. the Cacafuego.
Stephen ends up operating on himself in H.M.S. Surprise, then asks how long it took and is told "Twenty-three minutes just." His response? "Slow..."
Nothing beats Jack pressing his creditors' debt-collection enforcers early on in the series. "What's that? You're going to arrest me for debt? No actually I think you're going to work for me for the next year or so. Hope you don't suffer from seasickness!"
Unless it's Killick showing a camel his mettle in Treason's Harbour.No one steals the captain's personal effects. Becomes a Brick Joke later on, when the lengths that Killick goes to in order to safeguard the all-important chelengk end up foiling would-be spies from rifling the captain's personal documents.
Diana Villiers later Maturin has her own moment, when she gives away her most precious jewel to win the freedom of Stephen and Jack from French imprisonment.
More of Stephen's surgery: he very casually mentions that he had to amputate his own toes with a chisel after suffering frostbite. Damn, that's badass.
How Stephen finally deals with Wray and Ledward, the traitors who almost destroyed his marriage and Jack's naval career. This may also qualify as a moment of Squick, though it's mainly offstage: Stephen kills both villains and dissects their corpses with the aid of a Dutch naturalist whom he's befriended.