"They are called dogwatches note the dogwatch is only 2 hours long instead of the usual 4 because they are curtailed." An in-universe CMOF; it's still being referenced toward the end of the series.
After a day-long running battle a French ship is sunk by Jack Aubrey. The French officers are entertained by Jack & Stephen at dinner that night. When they are served spoiled, burnt soup it is treated as far worse torture than the battle. "Dipping their spoons they submitted to the horrors of war."
Or the time in Post Captain when Stephen came on board, wearing a truly bizarre bee-keeper's suit and carrying a hive of bees (which inevitably got loose, to the dismay of Jack and everyone else except Stephen.)
Or the time that Stephen adopted a wombat who ate Jack's best dress hat. Memorable enough that it gets a mention the next time they visit Australia, years and years later.
The description of Stephen and Diana's unconventional but successful domestic life in The Ionian Mission.
Any encounter Jack has with "Awkward" Davies, a brutish, unpleasant fellow who is not a particularly good seaman but who has an unbreakable claim on Jack's loyalty considering that Jack once saved the man from drowning.
Or the time that, when inspecting his crew, Jack found a seaman who had sold off literally almost all his kit to buy liquor and was down to one shoe. He unleashes a memorable smackdown on the unfortunate midshipman whose division the wretched fellow is in.
Then there's the time that Jack decided to quiz the midshipmen under his charge on their Biblical knowledge, specifically Abraham. Each answer he gets is more preposterous than the last, and the unlucky final middie gets a good tanning with a knotted rope's end from Bonden for his impertinence.
The first time we see Stephen let out a full-bellied laugh (or his equivalent, anyway; it's described as a dry, creaking sound) is when Jack informs him that he intends to preach a sermon to the crew at the next Sunday divine service.
Jack's opinion on Odysseus.
Jack explaining what a wad was to Clarissa Oakes.
Another one from Post Captain, "Jacques Aubrey"'s fake French identity papers, which are pure Refuge in Audacity. "Born 1 April 1066 at Bedlam, London. Father's profession, monk: mother's, nun. Mother's maiden name, Borgia, Lucrece." Sadly, the speaker gets cut off before he can get to "Etienne Maturin".
In The Fortune of War, Aubrey is wounded, and he and Maturin are captured by the Americans and taken to Boston. Maturin asks an American surgeon about improved accommodations for Aubrey, and the surgeon recommends a hospital that sounds excellent but is used mainly as a mad-house because of its owner's unpopular opinions and penchant for hiring Irish help. His remarks include some well-meant but highly unflattering statements about the Irish, right to the face of the half-Irish Maturin, until a semi-conscious Jack blurts out, "Never mind Maturin, he is an Irish Papist himself, ha ha ha! Drunk as a Lord every morning by nine o' clock, and never a shoe to his name." The surgeon is so mortified and apologetic that even Maturin's oft-dangerous temper is never roused.
While recovering in the mad-house mentioned above, Jack amuses himself by listening to the insane ramblings of his fellow-patients and telling them wild tall tales in return. When an American interrogation team comes to question him, their leader is an odd-looking man with an unusual name, so Jack mistakes them for more lunatics, and makes bizarre claims and confesses to outrageous crimes to the point where the interrogators think he is drunk.
Most sequences where Stephen has to get onto a ship would most likely involve him falling off a ship in one hilarious way or the other. It's a testament to their friendship that Stephen still agrees to volunteer to be Jack's ship surgeon after so many falls.
In The Letter of Marque, Jack trains the crew of the Surprise for a lightning harbour raid on the French coast, in a secluded bay. Naturally, this includes midnight boarding drills and gunfire. The reactions of the civilian dwellers on shore—many of whom had never seen a man-of-war—are hilarious, some mistaking the Surprise for a pirate ship come to carry them off. Even better, a revenue cutter, suspecting them of smuggling, tries to investigate but gets caught on a sandbar, and has to be towed off by Aubrey's crew.
When their ship is captured by a Spanish frigate who insists they surrender and be boarded, Stephen cheerfully asks if they have anyone who knows how to deal with the plague, because nobody on his ship has a clue. The Spaniards are more than happy to let them go.
Babbington being promoted to acting-lieutenant, only to remorsefully admit to Jack and Stephen the truth about the fate of Stephen's rats.