Any of the (many) moments when Temeraire says something incredibly awkward without realizing. Case in point:
Temeraire: Oh, you mean whores.
And in the same conversation, after which Laurence decides he needs a drink.:
Temeraire: "I am merely curious now, if that is all there is in Dover. For Roland is too young for whores, is she not?"
The short story Feast or Famine (set between the first two books) that has been made available on her website, all of it.
Not to mention when, in Throne of Jade, Temeraire confesses this to Lawrence...who then cracks up laughing, much to Temeraire's indignation.
In Black Powder War, Laurence orders his crew to take a hot soak on medical advice (long story), forgetting that his command is gender intergrated. One look and some frantic bundling into a blanket later:
Laurence: "You must not bathe with the others."
Emily Roland: "Mustn't I?"
Laurence: "No, it is not suitable; you are beginning to be a young lady."
In the beginning of Empire of Ivory , Temeraire attempting to explain the Pythagorean Theorem to guests at a party.
When Laurence bids Granby goodbye before they leave for Africa, Granby only says this.
"I expect you will find us all to pieces, with half of England on fire, and Arkady and his lot celebrating in the ruins, roasting cows; it will be wonderful otherwise."
Near the end of the fourth Temeraire book, Empire of Ivory. The titular dragon has received an invitation to a tea party from his Evil Counterpart. This is the exchange that occurs:
Laurence: "There is nothing evidently insincere in it; perhaps she means it as a gesture of reconciliation."
Temeraire: "No, she does not. I am sure if I go, the tea will be very unpleasant, at least my tea will be, and I will have to drink it or look ill-mannered. Or she will make remarks which do not seem offensive, until I have gone away and thought them over; or she will try and have you murdered while I am not there: you are not to go anywhere without a guard, and if anyone tries to murder you, you must call for me very loud."
The Riley/Harcourt shipboardnuptuals earlier in the book proved rather comical as well. The ship's chaplain was only sober because one of Harcourt's colleagues bodily hauled him up to the dragondeck and set him under guard (not that it did any good; he was so rattled that he stumbled through the service even worse that he would have drunk). The bride did not bother finding a dress, and the groom did not realize she wouldn't, so she ended up taking her vows in trousers and a coat (recall this is the early 1800s). Finally, when the chaplain got to the whole "Speak Now or Forever Hold Your Peace" bit...
Lily (Literally sticking her nose in): "Mayn't I?"
Harcourt: "No, you may not!"
Lily: "*sigh* Very well, (turns to look at Riley) but if you are unpleasant to Catherine I will throw you in the ocean."
Maximus recovers enough from being sick that he gets a hankering for a midnight snack, but doesn't want to wake his captain, who's been losing sleep from worrying about him. So he tries to sneak off and grab a couple cows. Live cows. That belong to someone else. On the other side of town.
Just to clarify, Maximus is one of the largest dragons in the whole series, nearly twice Temeraire's weight even half-dead from consumption. And a dragon this size tries to be sneaky...
This bit from Victory of Eagles, as Jane is defending her decision to send for Laurence. "And you may leave off your coughing and your insinuations: if I wanted a man between my legs so badly, there is a campful of handsome young fellows outside, and I dare say I could find one out to oblige me, without going to such trouble."
In Tongues of Serpents we meet Kulingile, a runt dragon born of two heavy-weights. Initially his survival is in question, and no one expects much of him, until a few weeks later when we get the signs that he'll not only live, but be one of the biggest dragons in the world: His air sacs inflate while he's sleeping and he becomes lighter than air and starts floating away!
The same book also introduces us to a minor character, O'Dea, a convict. Imagine a grizzled convict with a flair for the dramatic, which extends to his writing when Laurence makes him a scribe.
Jane Roland's letters in Tongues of Serpents crack me up. Especially where she explains differences that came with her new rank.
In Crucible Of Gold, Temeraire is ruminating over the discovery that Celestials cannot breed with dragons apart from their lineage, and his previous unsuccessful attempts to sire an egg. He ponders the idea of the problem simply being the 'unremarkableness' of those he was with, which is all Iskierka needs to hear. This also coming after several chapters of obvious Belligerent Sexual Tension. Cue a normally thoughtful Temeraire stumbling about himself, and just giving up trying to stop her.
"You do not need to hint...but I will try with you now, if you like, and Maila can wait," [Iskierka] said, in a rather venomous tone.
"I was not hinting—" But Temeraire shook out his ruff, and hastily said, "—oh, never mind; very well,"
In Blood of TyrantsLaurence is lost during a bad storm, and washes ashore Japan with amnesia. He doesn't remember anything of his time with Temeraire, or being an aviator. This of course leads to hilarity. Upon Temeraire finally finding him this exchange takes place.
"And you need not look like that," the black dragon added, very coldly, "only because Laurence has been shipwrecked, and does not look his best at present. The Emperor adopted him, five years ago, and we are on our way to make a filial visit. He is a prince of China, and my captain."
"The devil I am," said Laurence.
Halfway through Blood of Tyrants, Temeraire invites Lung Qin Mei to attend dinner with him and the other dragons - partly because he wants to see her, but also because he's been asked to sire a new Celestial companion for the Crown Prince. However, things go from bad to worse; first Iskierka jealous at her own egg with Temeraire being passed over insults Mei and leaves, next Temeraire becomes humiliated by the lack of manners of his companions, who abruptly leave so that Temeraire can arrange affairs with Mei. Finally, once they've left, Mei reveals she's been learning English to talk with him, and as such, understood everything that was said during the dinner. Becomes a Crowning Moment of Heartwarming shortly afterwards though, as the two are more happy to see each other than anything else.
During an argument in Blood of Tyrants concerning the possibility of marriage, Lawrence finally nerves himself up enough to ask if Emily Roland is his daughter (with the non-so-subtle implication that he must wed her mother). Temeraire promptly reassures him that she is not, pointing out quite logically that he did not start sleeping with Jane Roland until Emily was nine and it simply did not work that way.
"But it does not signify," Temeraire added in tones of reproach, "for you have already asked her, and she wouldn't have you, because she is your commander."