If you don't even wibble at the end of Commodore, you may be a robot.
Every time Hornblower is reminded that people actually like him, and every time Bush worries about his health.
Hornblower worrying over Bush's condition in Flying Colours (in an inversion of their usual position).
A brief line in Hornblower and the Cargo of Rice notes that although Hornblower is expecting to be laughed at for pitching headfirst into the ship's boat, it's not the way of the Indefatigable to mock someone giving his all without shirking.
Trying to take Wellard's mind off his troubles after being unjustly beaten, and trying (though unsuccessfully) to save him from the captain a second time.
Including Hornblower in the expedition to take the fort out of something like brotherly affection.
Taking the first watch after the attack on the fort so that the men can get some sleep, even though he's ready to fall over himself.
Outright ordering Hornblower to drink, eat, and sleep.
Deciding to give Hornblower "every credit" in the report about the attack.
Bush's anger over Hornblower being put under pay stoppage after peace is signed, which has forced Hornblower to pawn his sword, sell his pea-jacket in a bitter winter, and become a professional whist-player in a gambling house—a hellish outrage indeed. Bush immediately offers Hornblower his own half-pay for the month despite the hardship it would cause himself, and while Hornblower refuses, knowing that Bush gives a damn about his situation brightens his world considerably.
The moment near the end when Hornblower finds Bush nearly dead from bloodloss.
Hornblower and the Hotspur
Because Hornblower can't afford to give his cabin any of the comfort or decoration that's expected, Bush and the petty officers quietly fit it out with with curtains and cushions made out of No. 8 sailcloth, stuffed with oakum, with roses painted by some talented sailor. Hornblower is thoroughly shocked and delighted and has to confront the reality that they actually like him. (Of course, he considers this to be a sign of poor judgment, but it's a nice moment all the same.)
Hornblower quietly adds Seaman Grimes' name to the list of those killed in the attack on the semaphore and battery after Grimes hangs himself for showing his cowardice—this will let Grimes' family get his back pay and believe that he died in the line of duty.
After Doughty hits a bully of a petty officer and puts himself in line for the rope, Hornblower manages to lay him a perfect escape route to an American warship. Naturally, he excoriates himself for this good deed afterwards (because he did it for selfish reasons) but going over it, still thinks he would do it all over again.
A tipsy Bush visiting Hornblower at home in after Hornblower knowingly removes the ship from a battle that would gain him a fortune in prize money to stop a French ship that was rushing to protect the would-be prizes. It turns out that none of the captains profited from it, thanks to some Admiralty red tape.
"So we didn't lose anything on account of your noble action. Noble—I've always wanted to tell you it was noble, sir."
The fact that Hotspur was in the battle at all is a retroactive one, when Cornwallis tells Hornblower that he put him in the task force specifically because Cornwallis knew that Hornblower had been unlucky in the matter of prize money and thought he'd earned some for all of the brilliance he'd shown on blockade duty. (He wasn't aware of the red tape.)
At the end of Hotspur, Admiral Cornwallis retires, which gives him three promotions to hand out: midshipman to lieutenant, lieutenant to commander, commander to captain. As he's retiring, he says he can make these choices with only the consideration of the good of the service, rather than influence or politics. Naturally, he chooses Hornblower. Just as naturally, Hornblower doesn't realize what Cornwallis is getting at until he spells it out and when he does, is utterly dumbfounded.
"Thank you sir. I can't—" Very true. He could not speak.
Hornblower TV series
In "Mutiny", Kennedy instructing and comforting Wellard after the latter has been whipped by Captain Sawyer. Remembering what Kennedy suffered in the earlier movies, it's quite nice to see him helping a middie in a similarly bad situation. It's a short scene, and the boy gets whipped again not long after, but still.
Horatio's promotion in "The Duchess and the Devil". His surprised smile is too endearing for words.
"The Duchess and the Devil": Horatio's parting words to the Duchess who is, in fact, an actress Katherine Cobham in which they profess their friendship when Hornblower's men honour his word by returning to prison with him.
Even though the entire scene is something of a Tear Jerker, Horatio telling Archie he won't survive without him.
After Pellew shoots a sadistic officer who is trying to kill Hornblower:
Master Bowles:[enthusiastically] Exceptionally fine shot, sir!... [remembering his rank] ...If I may say so. Captain Pellew: You may, Master Bowles. You may.
In "Loyalty", when Hornblower puts across his view to Admiral Pellew that it's better to earn loyalty from the sailors than to demand it, this discussion ensues:
Admiral Pellew: When I was a child, I spoke as a child, I understood as a child, I thought as a child, but when I became a man, I put away childish things. Hornblower: Sir? Admiral Pellew: You know, Hornblower, it is very hard for a father to see his children grow up.
And true to Hornblower's character, the admission of fatherly affection goes completely over his head.
Pellew's speech about "a life of adventure and adversity" to Horatio at the end of "The Wrong War".
"Thank you, Mr Bush. But I will share the watch with you." After the initial tension between Archie and Bush, it's sweet to see the tacit acknowledgement that they've earned each other's trust and respect.
The whole scene from "Retribution" when the dynamic trio of truly awesome lieutenants return from the suicide mission to which only Horatio was sent, but Archie and Lt. Bush decided to disobey Buckland's direct order and help him. Everybody is impressed and chuffed. First one back on the ship is Archie and young Midshipman Wellard enthusiastically shakes hands with him, accompanied by heartfelt "Mr Kennedy, well done, sir!" and Archie thanks him cordially. Aaww! Then Horatio comes aboard and Matthewsnote who has been a member of Hornblower's division since the Indefatigable, and he acted as his mentor in "Mutiny" (as long as an experienced boatswain can be mentor to a young and highly competent Lieutenant) salutes him adorably and almost-hugs him, grabbing his arms with both his hands and shaking them, while he smiles with apparent relief, pride and happiness. Aaww!! Bush is also welcomed affectionately, and even Styles, with whom he shared some animosity, salutes him and decides to be quiet for once. Bush then goes to Buckland to apologize and provide some feedback on his unenviable situation, which Buckland surprisingly accepts with good grace. He also, in a rare moment of insight, gives us assessment of Archie, Horatio and Bush: "You three, you’re so full of yourselves. And of each other." Horatio and Archie are seen in the back, standing very close to one another and talking quietly. Just. Pure. Heartwarmth.
The scene in "The Wrong War" when after Archie saves Horatio from being blown up with a bridge, Major Edrington tells Archie gently, "And just... take care of [Horatio], will you?" Obviously understanding that Horatio needs emotional support, big time, and that Archie is the friend who can give it to him.
Also, when Mariette asks Horatio is not leave her alone at night, he promises to stand guard. She wakes up to find he is asleep, but still at his post.
Horatio nursing Archie back to health in The Duchess and the Devil, especially since Archie was purposely trying to starve himself to death. When he's later seen happily sitting outside with Horatio in full uniform, it's just lovely.