Aubrey's concern for the heavily wounded middie, Lord Blakeney, whose broken arm must be amputated above the elbow. Jack gives him a book to read about Lord Nelson's military escapades while the lad is held up in his hammock after surgery, and it's quite clear Aubrey sees the same potential for greatness in the little chap as his own hero saw in him.
An In-Universe example when Jack tells his officers at the Captain's table the second, slightly more serious, anecdote about Lord Nelson.
Aubrey: The second time... The second time he told me a story, about how someone offered him a boat cloak on a cold night. And he said no, he didn't need it. That he was quite warm. His zeal for king and country kept him warm. [Maturin scoffs, but tries to hide it] Aubrey: I know it sounds absurd, and were it from another man, you'd cry out: "oh, what pitiful stuff" and dismiss it as mere enthusiasm. But with Nelson... you felt your heart glow.
Lord Blakeney brings Maturin an unknown species of beetle, knowing he would appreciate some specimen of local fauna as he was very disappointed they couldn't stop at the Galápagos.
When Nagle refuses to salute and barges past Hollom, he doesn't reckon on being immediately spotted and called out on his actions by Jack, who is quick to jump to the midshipman's defence.
Capt. Aubrey: You there! Stand fast! Master-at-arms, take that man below and clap him in irons.
Kind of a sad moment as well, but Hollom remarking that Blakeney was always kind to him. He kills himself, but not without thanking the one person who liked him.
After Doctor Stephen Maturin has been accidentally shot and is dying, with the only real hope of saving him getting him onto dry land where an operation can be attempted to remove the lead ball. It is reported that the enemy ship has been sighted, and they are catching her; Jack looks out after the ship he has been chasing for most of the film through his telescope, goes into his cabin and sees Stephen's abandoned cello, and finally goes to see Stephen in the sickbay. The scene changes to Stephen waking up while being carried on a stretcher, having landed on the Galápagos Islands. He smiles wanly at Jack through his fever and says, "please tell me this is not on my account." To which Jack replies, "not at all, I just needed to stretch my legs." Just to add to the heartwarmingness, it is set to the soundtrack of Corelli's beautiful "Adagio From Concerto Grosso".
After Maturin performs the self-surgery, we see not only Captain Howard (who wants to absolve himself of the doctor's death), but also Joe Plaice, whom Maturin had saved when most ships' surgeons would have given up hope.
When Aubrey tells Maturin that they'll be staying on land for a week before heading back to England to repair the Surprise, and forgoing their pursuit of the enemy ship. This allows Maturin to pursue naturalist studies of the island that he had wanted to do before, but Aubrey had refused because the mission came first:
Dr. Maturin: Jack, I fear you have burdened me with a debt I can never fully repay. Capt. Aubrey: Tosh. Name a shrub after me. Something prickly and hard to eradicate. Maturin: A shrub? Nonsense! I shall name a new species of tortoise after you: Testudo aubreii!
Later on, he does repay him, by abandoning almost all of his specimens and rushing back when he manages to stumble across the Acheron at the other side of the island.
Capt. Aubrey: I believe Mr. Hogg would be a good choice for sailing master, however... that will be your decision, Captain Pullings.