The intro to Puni Puni Poemi features the main heroine brutally killing magical girls from various other shows, including one scene where she snaps the neck of SailorMoon.
Twice in the Read or DieOVA the President of the United States, who is visibly based on George W. Bush, is shown visibly losing bladder control in response to frightening events going on around him. Just to add an extra bit of English to the insult, though, the dub gives him a Texan accent not unlike that of Bush. R.O.D. The TV repeated the pants wetting gag and as further insult, revealed his name to be "President Al Gore" without changing who he actually resembles. America is also frequently shown to be power-hungry yet completely ineffectual and bothersome on the world stage. Britain is revealed to be an effectual power-hungry nation that is willing to obliterate entire countries and practice mind control on the world's citizens in order to retain its prominence. China is shown to be brutal but requires stealing others technology to succeed. Japan is of course innocent and chaste.
Magical Project S, made in the mid-90s, depicted a President of the United States that looked like a Valentino caricature.
This has been a common theme that dates back to Blue in the original games, who Oak chastises for not giving his Pokémon the love they deserve. A similar revelation occurs to the next game's rival, Silver, whose Golbat does not evolve into the happiness-required Crobat until after said revelation.
In the same vein Kagurazaka, a one-shot Ditto Dueler from Yu-Gi-Oh! GX is meant to be a jab to the TCG players that try to win duels by "net-decking", copying a Championship-winning deck in the hopes of achieving a cheap, effortless victory.
As another "take that", when copying one such high-powered deck against Sho, who is supposed to be one of the weaker characters, Sho beats him handily and admits he was inspired by how Judai beat the deck's original user. Which is pretty much how the TCG goes, once someone figures out how to break an overpowered deck, everyone follows suit and the overpowered deck becomes an underplayed deck.
The second episode of Cromartie High School has the main character sitting in school, delivering a rant in voice-over about how whenever a manga is made into an anime fans of the manga "bitch about everything from character designs to the voice actors." It slowly devolves into a self-depreciating Take That at anime's budget-cutting habits, however—the character then realizes "Wait a minute—some anime this is! Nothing's moving!" followed by a reused animation from the first episode of a pencil dropping and the character saying "Oh, oh, wait, now it's moving!"
Taken a step further when that last line is abruptly interrupted by The Endof the introduction.
Of course, since this is a satire series, it's loaded with Take Thats, from Michael Jackson to corporate sponsorship naming rights. They're also not as insulting as normal Take Thats are. And as if to leverage their external criticisms, the show is also fond of Self-Deprecation, both towards the show itself, Koji Kumeta (the author) and Studio Shaft.
Naruto has one that's fairly subtle and obscure: Deidara notes that his art is not "pop art" but "superflat". You'll only notice that Take That if you know what superflat and remember that Deidara is evil and his "works of art" are in fact explosives.
Earlier chapters of the Keroro Gunsou manga feature swipes at numerous Japanese politicians.
They also take swipes at the last US president and former Vice President Al Gore.
Full Metal Panic! has Shinji's dad mentioned he's surprised a model AS replicated the arm-joint weakness present in the real machine, but then notes that it's probably because it's made by Banpresto.
The Code Geass fandom has a rather vocal segment which feels that the School FestivalFiller episodes are pointless wastes of time that distracts from the main plot. Bandai's response came in the dub, when during one of those selfsame School Festival episodes, Milly comments that "Sometimes life just has these little filler moments - and that's okay!".
Pluto makes only the barest attempts to hide its criticisms of the Iraq War. The United States of Thracia suspects another country of making Robots of Mass Destruction, but before other countries can complete their investigation into whether it's true, Thracia sends troops in and starts a horrific, wasteful war that devastates the country and leaves just about everyone who fought in it with PTSD, human and robot alike. And the president of Thracia is a guy taking orders from a teddy bear.
Basically the entire series is based on Shout outs, Spoofs and Take Thats.
In Maria†Holic, Mariya chastises Kanako for mistaking a cross necklace for a rosary, describes what an actual rosary is supposed to look like and be used for, and points out that they're not meant to be worn as jewellery — i.e. a series of swipes at Maria-sama ga Miteru. The manga also takes potshots at series like Ghost in the Shell and Sengoku Basara.
The dub of Ghost Stories has a few examples. Subtlety is for the weak: "Our president is an idiot and a liar. God bless America, land of free speech!"
In Azumanga Daioh, Chiyo's father has the Berserk Button of people not believing that he is a cat. However, he is especially angered when Osaka thinks that he looks like the former Prime Minister, Mori. As indicated in the link, Mori was really unpopular, to a near 0% Approval Rating extent. No wonder Chiyo's dad was pissed.
In the dub, Osaka suggests that he resembles "that American man, Bill Clinton".
Hunter × Hunter has some fairly blatant ones against the government of North Korea. The repressive Republic of East Gorteau is run by a guy whose name is an anagram of Kim Jong-Il, and he gets eaten by a giant ant shortly after being introduced (and after the giant ant makes a little speech about how pathetic the dictator is).
Go Nagai is responsible for one of the biggest take thats in the history of manga if not one of the biggest take thats in general. After PTA forced Shonen Jump to end his debutant series Shameless School, Go Nagai finished the series by having the titular school attacked by the PTA; all of the main characters died fighting the evil forces of censorship.
Not the only time Nagai fired back at criticism: upon being told to dial the contents of his works down because it was supposedly a negative influence on children, Go created The Abashiri Family, which had more violent and overtly sexual content than anything he had done before.
Team Medical Dragon is one massive Take That to the Japanese Medical System, there is generally more focus on the politics of the profession than the actual medicine. That makes the original writer's death from Liver Cancer complications all the more poignant.
In the English dub of Dragon Ball Kai, Goku references the CW turning Mr. Popo blue by pondering what the dragon on Namek will look like, adding "Maybe he's blue."
The Funimation dub of One Piece will sometimes go above and beyond even the original Japanese with their language and adult references (such as alcohol), some of the more pointed, out of the blue ones are very likely to be Take Thats towards 4Kids and their dub.
Of course, the whole "franchise-killing" bit is a false rumor. The real reason why Tomino made Victory Gundam the way it was, is due to being entirely fed up with the Executive Meddling his projects were always subjected to (as was the case with V Gundam itself at the start, naturally).
In Saki, one "Saki Biyori" side story has a Take That against horror movies that take a long time to get to the scary parts. Hiroko of Senriyama, having two such movies due the next day and not enough time to watch both, gives one about a zombie to Toki and Ryuuka, and another about a shark to Izumi and Cera. It takes over an hour and a half (out of two hours for the shark movie) for the monsters to show up, and until then, Toki falls asleep in Ryuuka's lap, while Izumi and Cera make small talk about Izumi's math class to pass time.
Bakuman。, being a series about writing a manga that occasionally references real manga, has a few.
An interesting example to fans of Death Note: When Mashiro and Takagi start attending college, they mention how good it is for them to have Miyoshi hang out with them due to how they'd appear to be a couple if they didn't have a girl with them (Mashiro has a girlfriend, and is technically engaged to her, but this is a secret to which only a handful of their friends are privy). In other words, Ohba and Obata are quite aware of the gay subtext made between Light and L when they start attending To-Oh Uni together, and their apparent closeness (in spite of L trying to prove that Light is Kira, and Light trying to find a way to get L's real name and kill him without incriminating himself). Yes indeed, the fans were rumbled.
Toward the end of the series, after Mashiro and Azuki's relationship has come to light, imperiling her chances of getting the role in his anime, Fukuda shares his thoughts on the radio.
Fukuda: The voice actress you like has a boyfriend... Sure, I understand that some of you would be depressed to hear that. That's the fan spirit. (yelling) But even so... using that as an excuse to post all kinds of stupid crap on the Internet is something I can never forgive! Voice actresses and manga artists are humans, too. There's nothing wrong with them having a relationship!