The dinner that Captain Cogshill hosts near the end, after he takes command and the court of inquiry concludes that there is no need for a trial. Cogshill then proceeds to get himself, Bush, and Buckland absolutely sloshed with half the drinks brewed in Europe. Bush's mood improves with each successive toast (such as "Glorious war, oceans of gore, prizes galore, and beauty ashore"). The text notes that for the rest of his life, Bush remembers it as "one of the most successful dinners he ever attended."
The academic debate on the Natividad's firing between Bush and another lieutenant. Specifically, they're discussing the way that the Natividad is shooting at them at this moment—they are briefly interrupted by a need to give orders regarding damage, and then Bush seizes on that as evidence for his opinion.
When first introduced in Beat to Quarters, Hornblower has fallen back on clearing his throat ("Ha-hm") to avoid being over-talkative to his officers. Lady Barbara promptly teases him into abandoning this "useful non-commital sound".
Lady Barbara: And if my name should escape you, Captain, you can always simply attract my attention by saying "Ha-hm."
In one of the books, Hornblower orders the men into action, and some nameless wit shouts "Ol' Horny is at it again!" The officers try and fail to figure out who said it.
In Ship Of The Line, Captain Hornblower joins a Marine officer to observe an ongoing ground battle. The Marine reassures Hornblower that the French marksmen have no hope of hitting the two officers at this range... and Hornblower's hat is promptly knocked off of his head by a musket ball. Hornblower decides that it is time for him to retire from this battlefield, and the other officer agrees.
Commodore Hornblower has to quickly come up with an excuse to get out of the ceremonies for the Czar so he can stop an assassination attempt on the Czar. So what does he go with? A Potty Emergency.
Admiral Hornblower discovers to his dismay in West Indies that the Governour's Daughter has fallen in love with him with the obsessive tendency that only a teenage girl can muster. Through the course of the story, he manages to redirect her affections towards one of his lieutenants, who being both young and unmarried, is much more appreciative of the attention.
Captain Horatio Hornblower R.N.
Bush and one of the Lydia's other lieutenants make a bet—Bush knows exactly what the captain will say to Lady Barbara when she comes aboard. Cut to Hornblower greeting her with an incomprehensible, awkward cough, then back to Bush with a very satisfied look on his face.
Hornblower TV series
From "Retribution": I CAN'T SWIM! Shouted by a panicked Lieutenant Bush as Hornblower and Archie basically drag him off of a cliff with them to avoid getting killed by an exploding Spanish fort and being stranded on an island held by the slave rebel army. For those wondering about an officer of the Navy who cannot swim, read an article on Wooden Ships and Iron Men. It was a fairly common thing. Later on, when Archie questions his own bravery, Hornblower points out that he jumped off a cliff into the ocean accompanied by "A man afraid of heights and another who couldn't swim." Which would be funny if Archie hadn't been dying.
Archie's overly-interested reaction to the officer and lady kissing on the tower in "Retribution". Bush is not impressed.
"Retribution": Hornblower is blowing up a Spanish fort, when Kennedy and Bush turn up to join him.
Horatio: Archie, are you mad?
Archie: Very possibly, but we thought you could use the company!
Any moment when Captain Pellew chews out Hornblower for some reason or other... then hands him a promotion or a new command. Horatio's stunned expression is priceless each time.
In "The Wrong War", Hornblower's failed attempt to get on a horse. He ends up going around in circles, one foot in a stirrup, much to Major Edrington's amusement.
Edrington: I can see why you chose the Navy.
From "The Wrong War"/"The Frogs and the Lobsters", Horatio and his riding skills:
Horatio: Damn this animal. What it needs is a rudder!
Edrington: Show it who's master, Mr Hornblower.
Horatio: [desperately] I don't need to, it knows.
The fantastically awkward moment in Series 2 ("Mutiny") where Horatio is showering on deck, and Captain Sawyer comes out of his cabin to arrest his lieutenants. The moment where Horatio realises why everyone has suddenly stopped and what they're looking at is immediately followed by his look of horror at having been caught prancing around starkers by his lunatic captain.
The captains on the board of examination in "The Fire Ships". After Hornblower and "Dreadnought" Foster have stopped the titular ship from destroying the fleet at anchor, Foster and Hammond immediately start bickering and get so riled up that Hammond challenges Foster to a duel. Hornblower's bewildered and slightly hypothermic expression as he watches them makes the scene all the better. Deadpan Snarker Pellew later describes them as "three men who would not agree on the color of an orange".
Horatio's Tactful Translation to a Spanish captain who brought them a message about Spaniards' becoming neutral. Pellew is furious and tries extremely hard not to let his mask of politeness slip and gives a crooked smile, while Horatio says random polite stuff in French. The unassuming Spaniard is impressed and even bows in respect.
This fan video has a pretty solid selection of priceless moments from "Mutiny", "Retribution", "Loyalty" and "Duty". Some of them were funny intentionally, some of them... not so much.
#1 The Tackle: Mr Bush comes aboard the Renown, and is nearly hit by a net full of barrels. Horatio knocks him down just in time. Interesting welcoming ceremony, indeed.
#2 The Two-syllable There: Mr Bush suggests to Horatio that something more than met the eye happened when captain Sawyer fell.
Mr Bush: Is that all?
Mr Hornblower: All? What do you mean all?
Mr Bush: You know what I mean. You were the-ere.
#3 Quietly as You Can: A disagreement between Styles and Mr Bush who suggests that if Styles is loud during their night attack, he will silence him for good. Matthews, despite being friends with Styles, approves of Mr Bush's interference, and then not-quite-whispers (that is, he shouts) to the crew: "Softly now, quietly as you can."
#4 Gawking at the Lovebirds: The already mentioned scene. Archie obviously likes to watch. Who knew he was such a voyeur? And how come a voyeur can be so adorable?
#6 Indignant Horatio: Features Horatio's Shot vs Mr Bush's Shot. Horatio is not pleased that his was a miss.
#7 Sourpuss Horatio: Horatio tries using heated shots to attack the Spanish ships. Unfortunately, one of the guns overheats and blows. Chaos ensues. Luckily, their crew manage to run and they do not get blown up as well.
Mr Bush: I know what happens when metal overheats. My uncle was a blacksmith.
Mr Hornblower: I believe they call it the co-efficient expansion, sir.
Mr Bush: Oh. I call it bloody dangerous. [Mr Kennedy gives a laugh.]
Mr Hornblower: I'm glad you find some amusement in this situation, Archie. Because of my elementary mistake...
Mr Bush: Come, gentlemen, no time for soul-searching. The furnace awaits. [They leave, and Horatio gives them a hard look which is very close to a stink eye or even a Death Glare.]
Mr Kennedy: [being saucy] I remember when you used to be scared of heights, Mr Hornblower.
Mr Hornblower: [measures the height he must descend from] Nothing's changed, Mr Kennedy.
Mr Bush: They say one must always do what one dislikes.
Mr Hornblower: Oh, yes?
Mr Bush: [nods] When I was a boy, I had to eat turnips.
Mr Hornblower: Eat them now, do you? [starts to climb down]
Mr Bush: Never touch 'em. [smiles]
#9 The Jump: I caaan't swiiiim! See above.
#10 The Spanish Ladies: Mr Hobbs desperately wants Captain Sawyer to remember his accident and leads him to the hole where it happened. Ominous music keeps playing. When Hobbs asks what he can see, Sawyer answers: "Spanish Ladies." True enough, in the very place are now kept imprisoned Spanish Ladies.
#11 The Back of Mine: Mr Bush is a bad ass 1st Lieutenant, and is not afraid to bitch-slap.
Mr Bush: [to midshipmen Hammond and Orrock] As Midshipmen, I expect you to know this ship like the back of your hand, or else you shall know the back of mine. Is that understood?
#12 The Voice Crack: Mr Bush's voice breaks when he orders the crew to follow him to attack the fort:
Mr Bush: Renown's, to me-e!
#13 Funny French Face: Major Côtard played by Greg Wise has priceless facial expressions.
#17 Horatio's Flood Pants: Too short pants and white socks? While climbing a dark rock? Not a good fashion choice, Horatio. What happened to your nice uniform?
#18 The Horrifically Butchered American Accent: Some sensitive ears don't take lightly Fake Americans.
#19 Like True British Sailors: Crazy Captain Sawyer is doped by laudanum (it must be a lethal dosage by this point) and is totally losing his wits. When Mr Buckland hosts Colonel Ortega and his wife so that those would lose their attention, while the other lieutenants prepare for action, Sawyer decides he wants to sing as loudly as possible. The uncomfortable looks, especially Senora Ortega's, are just hilarious.
#20 The Pity: Captain Sawyer singing performance continues and Acting Captain comes to his room to silence him.
Acting Captain Buckland: Sir! Stop this madness!
Captain Sawyer: Ah. The acting captain. So glad you're here.
Acting Captain Buckland: Sir?
Captain Sawyer: When I see you, I feel so much less pity for myself.
#21 The Mildew-ed Canvas Bed: Major Côtard who is by this point of the story just insufferable French Jerk does not like his berth. Or berth-mate?
Major Côtard: [absolutely outraged storming into Captain's room] Capitaine 'ornblower, if this is a joke, it is in poor taste.
1st Lt. Mr Bush: Sir, only Captain's steward may enter without knocking.
Captain Hornblower: What's the problem, Major Côtard?
Major Côtard: Do you seriously expect me to sleep in a mildewed canvas bed with a common crew?
Captain Hornblower: [to Mr Bush] I thought he was sharing your berth?
1st Lt. Mr Bush: He is.
From The Even Chance, Captain Keene is overseeing the Midshipmen's training, and is checking their work. What follows is a long series of him checking their slates and saying "No... no... no..." as each one proves incorrect. Judging by one midshipman's face, he genuinely was proud of the answer he presented until the Captain shot him down as wholly incorrect.