The anime Psycho-Pass has a stealth pun in its title. In Japanese, 'Psycho-Pass' and 'Psychopath' sound the same. Considering what sort of society this anime is set in, and what type of people inhabit it, both words would work equally well as the title.
Beyblade Zero G has two characters, Shinobu Hiryuin and Eight Unabara, whose Beyblades - Shinobi Saramanda and Pirates Orojya, respectively - can be Synchromed (as in combined) with the Beyblades of the two main characters (Zyro Kurogane and Kite Unabara), making the two indirect rivals of sorts. As for the Beyblades themselves... Pirates vs Ninjas?
In Magical Girl Lyrical Nanoha, Nanoha is pursuing Jewel Seeds. One such Jewel Seed had possessed a tree that was near a couple's confession of love, and it responded by turning into a massive murdering monster, trapping them inside itself and trying to consume everything. The pun comes when you realise that they pined for each other. (This pun actually works in Japanese, too.)
In Chobits, Hideki is searching desperately for the on switch for his new discovery, a female robot, when he says there's only one place left he hadn't checked. Cut outside to a mewing cat.
Ooh, there's a double there, too, at least in English...
The main character of One Piece is named Monkey D. Luffy, and his first appearance in the anime was breaking out of a small barrel. Perhaps the implication is that the show will be more fun than a barrel of monkeys.
Another pair are for Mr. 1, his devil fruit is called the Supa-Supa fruit, it gives him super powers and his secret dream is to be a super hero. The obvious one is "super" a play on "Supa", his devil fruit name. The second not quite pun is he is literally a Man of Steel. A "Supaman" if you will..
Later in the series, Sanji uses a move called the "Parage Shot" to change people's looks by kicking them in the face. In other words, he can literally "rearrange your face".
Bepo is a human sentient bear who goes from a vicious kung fu brawler to an apologetic depressive the next. In short, he's a bipolar bear.
Arlong is a great white shark fish man who used an impossible loan to enslave Nami. Truly a loan sharks for the ages.
Greed claims both that he wants everything in the word and that he never lies. Thus he always keeps his word.
A particularly painful one is the Mobile Suit Gundam MSV model MS-07H-4 Gouf Flight Test Type, a prototype Zeon mobile suit adorned with their iconic mono-eye camera and forehead-mounted communications antenna, and painted in outrageous magenta demo-colors. An early, unsuccessful attempt at creating a flight-capable Humongous Mecha, it was scrapped for its unfortunate tendency to kill its test pilots. Long story short, it's a one eyed, one horned, flying purple people-eater.
In another, more obscure Gundam example, one of the various artbooks features a scene with Haman Karn eating a piece of cake with a candy on top shaped like one of her Humongous Mecha Quebeley's funnels.
At the end of the first episode of Gankutsuou, Albert is making out with Peppo, the local "Bridget" (although he doesn't know that yet), when she stops him and, points a gun at him, and helps kidnappers take him hostage (thus, she had a pistol in her pocket, but might have been happy to see him as well).
There's one that's true of both RahXephon and Seikimatsu Occult Gakuin. Both series make reference to the whole "2012 End of the World" idea, and it's probably no coincidence that both also have characters named Maya.
Charles Goodshow is the president of the Pokémon League and appears in the episodes pertaining to the larger tournaments. He is a short elderly man with a large beard, bright-colored shorts and a backwards cap, giving the visible impression that he is still a child at heart. Goodshow, old boy?
In New Getter Robo, there's a pun in the first episode. When Ryouma is told by Professor Saotome that he is to be one of the pilots of the Getter Robo, Ryoma replies "Getter?" With the scene being shot at floor level, focusing on Saotome's "geta".
A flashback in Naruto has someone hearing what Sasuke's name is, and commenting that's also the name of the Third Hokage's father. The Third Hokage's real name is Hiruzen Sarutobi, thus making his father...
During one of the filler episodes, Naruto has to sit at a funeral and not laugh the entire time, or else the person they're doing the mission for will not get the inheritance. So naturally everyone is trying to make him laugh the entire time. One of the gags involved a guy having a flower coming out of his nose. In Japanese, the word 'hana' can mean either "nose" or "flower".
Hidamari Sketch: When the residents of the Hidamari Apartments go to a sentō following the cultural festival, they relate things that they did and experienced. For example, during a Cinderella play they put on, Miyako did the voice of a horse; while in the baths, Hiro asks her to replicate it. Miyako refuses, not wanting to sell herself cheap, but an unspoken reason is that horseplay is against bathhouse etiquette. (Probably not intended...)
The Tatami Galaxy represents the protagonist's libido via a cowboy with a phallic-looking nose called "Johnny". The intent becomes clearer when you realize that he looks a whole lot like Woody from Toy Story.
In the Virtual World arc in Yu-Gi-Oh!, Noah, the arc villain chooses Shinato's Ark for his Deckmaster. It is never mentioned that this would make it Noah's Ark.
Heck, that pun is used twice; fans tend to refer to that whole arc as Noah's Arc.
In Black Butler, demonic butler Sebastian Michaelis' catchphrase is "I'm just one hell of a butler." Has the bonus of working perfectly in its original language as well.
To be clear, since the Japanese language can differ in meaning based on subtle inflection, Sebastian's catchphrase is either "I am a devil and a butler" or the more subtle "I am a devilishly good butler."
The title of Neko Kissa is either a Bilingual Stealth Pun or a Accidental Pun. In Japanese, "Neko" = "Cat" and "Kissa" = "Cafe". So, "Cat Cafe". The title could also be translated as being the word "cat" in two different languages - Japanese and Finnish: Neko (Japanese), Kissa (Finnish).
In Accel World, Haruyuki's normal neural linker avatar is a pig. Silver Crow's main power is flight. Therefore, Pigs are flying.
Shaman King: Lyserg Diethyl, one of Yoh's allies who eventually joins the X-Laws. People from the 60's or familiar with the Hippies may be able to pick it up, but it's a reference to Lysergic Acid Diethylamide...aka LSD, infamous hallucinogen that supposedly let you talk with spirits. Spirits are the whole premise of the series.
Then there's his spirit partner Morphine, if you want to get really blatant.
Code Geass: Suzaku was Private 404 in the army. Given his status for most of the series, it fits.
Here's a much subtler one. Lelouch's Geass allows him to command anyone to do anything, but it only works once per person, and it can't be undone, which means that they'll still be doing that command until they die, or get un-Geassed by someone with an ability to do so. Lelouch Geassed Suzaku commanding him to "Live." Given the nature of his Geass, Suzaku's gonna Live until he dies.
In Toriko, whenever the Nightmare Heracles, a horse with godlike powers, gives birth, solid rainbow bridges appear that allow her offspring to run over to other continents. In other words, we have ponies doing literal rainbow dashes.