Neon Genesis Evangelion plays with this, as the dub can really be hit or miss. On one hand, you have lines like "Every single missile hit the target!", but the last two episodes are captivating in their Mind Screw.
Lots of the dramatic scenes in One Piece are incredibly over the top even for manga. Because the rest of the story is just as exaggerated and fantastic, and the characters and situations themselves are uniquely compelling, there's still a rather moving effect. All the snot, tears, and loud broken groaning are usually indicative of Narm (even the girls had them), yet here it makes the story more raw and emotional.
The Narm from Trigun's Vash and Knives is entirely appropriate for their childlike, naive personalities. Vash is an idealist who honestly believes that all the evil and suffering of the world can be conquered by love and peace, takes harm on himself to spare others, and wants nothing more than for people to stop hurting each other. Meanwhile, Knives can casually kill anybody that he pleases — including his brother whom he actually does love — but collapses in incredible fear and confusion when he himself is hurt. This arguably makes Knives' psychopathy that much more terrifying, hating pain but happily inflicting it on everybody he considers "beneath" him, as though they were mere insects undeveloped enough to experience pain themselves.
Many other scenes already described under Narm also tend to have the Narm Charm effect, such as Euphemia's smiling face and cute personality mannerisms even while ordering the deaths of Japanese people near the end of the first season.
Everyone makes fun of the infamous potato-chip scene in Death Note, or the somewhat "overexpressive" anime in itself. Still, it is a very popular and critically acclaimed show with a ton of fans.
This is especially evident in the final episode, in which the over-the-top fashion in which Light and Mikami break down runs a weird line between comical and horrifying.
The official English translation of the Mahou Sensei Negima! manga has some rough spots, but it's also responsible for several great lines from Negi, such as asking if he can have a cookie after getting wedged between Shizuna's boobs, and yelling "Damn my charisma!" upon getting mobbed by a group of high school girls who think he's the cutest thing ever.
Evangeline's: "They who have the most guns, kick butt" and "Are you on CRACK?" note Her reaction when Negi asks her if he can be her disciple.
In Pokémon: The First Movie, we hear the two Dewgong crying in their ridiculously comical voices. But the fact that they're crying over Ash being killed stops it from being funny. It also helps that the next shot is of the Vulpix crying which is a genuinely sad sound.
In the same film, a lot of Mewtwo's dialogue sounds comically overwrought and stereotypically villainous if you walk in after the start of the movie, which shows his seriously sad childhood and growing distrust of humans.
The way Goku strains when trying to power-up or endure a lot of damage inflicted upon him.
Any of Frieza's "Am I male or female?" quotes.
"Having these balls gives me something that resembles joy, I think. I want to caress them"
In his first form, Japanese Cell frequently makes a strange "Bwaaaah" sound. It's a silly sound, but Norio Wakamoto makes it sound chilling.
And that bizarre sound he makes when walking. It makes it seem like he has shock absorbers in his legs or something, but considering he's an engineered life form it fits.
The second episode of Darker Than Black has a line where Mao comments that the protagonist's coat is "bulletproof, but only when he wears it." This is simultaneously inexplicable and really badass. This line was altered in the dub to be a statement that the coat is "not just a fashion statement".
And then there's the ever so memetic NOW I'VE LOST IT/I KNOW I CAN KILL mantra just before the OP kicks in. Ridiculous? Maybe, but you're probably too busy gleefully chanting along to care.
In the beginning of the second season, as Suou narrates about how the world has changed since the Gate appeared, a very futuristic flying car briefly appears on screen. Kind of narmy because of how much of a cliche flying cars are as a symbol of the future, but it kind of works because weird Magi Tech and Black Box technology is a big part of the setting. And the fact that flying cars are just cool.
The scene in season 2 where Ilya talks to/scares those tourists might qualify. It's really hard to tell if that moment was meant to be somewhat humorous or not. If it was meant as a serious moment, they failed, but it works well as Black Comedy.
In Gundam 00, during the original Lockon Stratos's death scene, the little robot Haro crying "Lockon! Lockon!" was a tad bit narmy. At the same time, though, it was rather effective. I mean, come on, even the robot was sad!
In the first episode of the second Yu-Gi-Oh! series, Yugi summoning Exodia is narm charm enough. Then watch the uncut video, especially past 3:00.
The dub dialogue is on fire, helped by Dan Green's delicacy...
Kaiba: "Draw your last pathetic card so I can end this game!"
Yami: "My deck has no pathetic cards, Kaiba. But it does contain... the unstoppable Exodia!"
Kaiba: "Exodia? Impossible! No-one's been able to summon him!" note Yeah, there wasn't a whole lot of search back then. Shut up.
Yami: "Exodia, OBLITERATE!"
Really, just the whole first series in general is this, not just the dub. Sure, people getting so dramatic about card games is hilarious, but you're often too busy crying or cheering to care.
Just Yu-Gi-Oh!, in general. Even before the card game was introduced, the whole thing was so spectacularly, unashamedly over-the-top that it was difficult not to love.
The dub of the sequel series Yu-Gi-Oh! GX generally consists of general old Narm. One exception is when a character steals Yugi's deck and begins imitating his voice. We get an voice actor who is usually a Large Ham imitate Dan Green. And it's awesome.
In the second sequel series Yu-Gi-Oh! 5Ds, the English dub always has to remove stuff. Naturally, they edit out weapons, text, add horrible puns to what should be relatively serious dialogue, but the main reason the dub is decent is simply because the Narm is so goddamn concentrated that it must be intentional. In one scene where the main character Yusei is in a motorcycle accident, when his friends go to help him he visibly has shrapnel sticking out of his chest. But not in the dub, no.They edited it out, and added the cheesy "Augh, my gut!" line instead. They turned what was supposed to be internal bleeding into "Thanks, Pepto Bismol!"
Also, because of the habit of ignoring most FATAL things a lot of issues were changed. Even though, by the time of a certain episode beginning to explain the plot, the season finale has occurred in Japan. And only 2 people STAY dead. Rex and Rudger/Roman Go(o)dwin. So to protect the children, the plot got so warped that they got a new "Shadow Realm" in "The Underworld."
Sailor Moon has a good example of this. One episode had Nephrite's death, and the line that comes next has Narm written all over it: "I'm so sorry, Naru... I guess I won't be taking you out for a chocolate parfait." It Makes Sense in Context, but still sounds silly when taken by itself. No one cared because they were sniffling...
There is plenty of Narm Charm in the dub: After apparently escaping Zoicite and her mooks Molly mentions the local Cafe makes exquisite Cholote Parfaits and asks if Nephrite if they could go have one sometime,which he agrees to. Then she asks him if he has "Holidays in that Evil Society of His." The sheer silliness of the second question,already silly in Japanese, gets funnier with Molly's goofy Brooklyn Accent, Nephrite and Molly herself have a bit of a laugh about it. Instead of ruining the scene, it somehow manages to keep the drama and pathos of the Japanese version wonderfully.
JoJo's Bizarre Adventure. From the FAAABULOUS character designs, to the outrageous abilities (fighting with a bladed hat, putting zippers everywhere, turning rocks into mud, stopping time), to the characters themselves... then there are the manly tears, such as when Polnareff cried when Abdul took a bullet for him, to when Joeseph saw his daughter's stand turning on her, et cetera. Yet, somehow, Jojo manages to work.
The next thing you're going to say is ____.
Muda da, MudaMudaMudaMudaMudaMudaMudaMuda! vs OraOraOraOraOraOraOraOraOra!
Road roller/Tank lorry da!
Detective Conan: Conan's use of broken English while telling his former idol, Ray Curtis that his circumstances do not justify turning to drugs and murder would otherwise be funny, but it, combined with the music and his facial expressions, effectively conveys how the poor guy feels, and Ray Curtis's response in perfect English makes it clear that the grammar mistakes are Conan's, rather than the writer's.
Zoids. The first series (Zoids: Chaotic Century) in particular. The entire script is so laughably bad it's amazing that everyone keeps a straight face. (Van and Raven's exchange in episode 33 is a personal favorite example) But that's the reason it works. The whole universe speaks in narm and once you get used to it you realize that it's the language of the universe and it's a much more entertaining universe as a result.
Eureka Seven, episode 48. Anemone is on a suicide mission after having learned that Dominic has abandoned the military to stop the Big Bad's plan. Lines like "I want to live! I wish I didn't have feelings like this!" are tearfully screamed at full blast, with all the pretentious angst of a bad Goth band. note This makes sense, given how this character is a severely emotionally crippled and confused teenager. The viewers are just as likely to be crying themselves at this point. note If not, they're probably saving up the tears for the happy reunion of Dominic and Anemone a few minutes later.
The entire PREMISE is built on this. The series is built on giant surfboarding robots, tons and tons of blatant counter-culture references (including one of the most disastrous incidents in recent history being called the Summer of Love), and an incredibly sappy romance at the forefront (hell, the main mecha of the series is practically POWERED by love.)
During Itsuki's rooftop talk with Kyon, after Haruhi has fallen asleep beside Mikuru, he suggests to Kyon that he should "embrace Suzumiya-san from behind, and whisper 'I love you' in her ear." What the English audience hears is assorted Japanese, and then, "AI LAAV YU." What keeps it from falling into Narm is the heart-warming playfulness of the scene.
The Image Songs use a hefty pile of Grautitous English in their lyrics. Lost My Music has a chorus in English, while God Knows... appends English words to the ends of certain verses.
Christy Vee's translation of "Hare Hare Yukai" has her first line shouting (in the most high-pitched voice she can muster) "ALL RIGHT SOS BRIGADE, ASSEMBLE!" It's incredibly cheesy, but also incredibly hilarious and in-character for Haruhi.
When the ending of Umineko no Naku Koro ni's second arc was adapted to anime, quite a lot of people were collapsing in laughing fits over the "takomaria". However, there was a significant creepy factor to seeing her head rolling around like that, and in a series of over-the-top messed-up scenes like the banquet of the witch and Rosa's feast afterwards, to a certain extent, it actually fit pretty well.
OH DESIAH! In spite of the Narmy Engrish — or perhaps because of it—the song still manages to be epic.
In Higurashi no Naku Koro ni Satoko's breakdown in episode 10 involves her somehow throwing a guy twice her size all the way across the room, with the chair he's sitting on. The voice actor's delivery, combined with the music, is what makes the scene work.
Fullmetal Alchemist. Father's expressions are often over the top, but he's still the main source of Nightmare Fuel of all the series... especially when he looked pregnant of Hohenheim after having just absorbed him... but his One-Winged Angel form was absolutely terrifying.
Episode 167 of Naruto Shippuden. Full stop. It goes into full-on Looney Toons action at various points, including the "hammer person into ground without killing them" and "Road Runner legs" effects. Yet, it can also be absolutely awesome.
Itachi's epic freak out may be ridiculous in the manga though since it was an act, this may have been the point; it's also seemingly the terrifying instant when a normally emotionless, albeit already evil, character reveals his true nature as a complete and utter psychopath. In the anime, both Hideo Ishikawa and Crispin Freeman's respective performances are so utterly bone-chilling that the scene becomes much less ridiculous than it was in the manga.
The Intelligent Devices of the Lyrical Nanoha series have always spoken in rather questionably-translated English ("It can be done. As for my master.") Then The Movie came along and gave them proper English lines. Many English-speaking fans are not pleased with this, as they found the Devices' broken speech endearing.
Hellsing's hamminess and over-the-top violence should be ridiculous, but many fans love the series because of it.
Belldandy from Ah! My Goddess continues to call Keiichi "Mister Keiichi" even tough they have lived together for a pretty long time. And she is just an overly melodramatic sugarqueen in general. But you just can't hold it against her when it comes with that weapons-grade smile.
Many aspects of the show count as Narm Charm—the relationship between the two leads, the frequent use of Japanese Stock Phrases, the campy costumes and the theme of friendship being noticable examples. Tiger & Bunny 's particular flavor of Narm Charm is actually rather simililar to that of Code Geass—both shows contain seemingly ridiculous amounts of drama and camp, but manage to achieve some really touching moments regardless.
The Engrish opening and ending themes to the Berserk anime manage to be both hilarious on account of their mangled pronunciation and borderline-nonsensical lyrics (and a completely straight-faced use of the line "I can get it off!") and awesome on account of being damn good songs regardless.
The Manga Bible; as noted in World of Ham article, the characters are so dramatic, the anachronism so blatant (calling someone "punk" in 1500 BC, think about it), but it still manages to keep the Original Flavor from the source while maintaining it as entertaining reading (for everyone, averting the Confirmation Bias trope).
Demon King Daimao has Kena Soga, who, depending on the dub version, either has a certain charm to how her voiced is portrayed. Aki Toyosaki has a more wacky and cartoony route for her, while Melissa Davis makes her overly-dramatic and goofy.
The fight between Masako and Yuri in Episode 15 is so ridiculously over-the-top (it involves Yuri swinging ping-pong paddles to swat Masako's slingshot balls, for crying out loud!) that it warps back to ridiculously awesome.
The whole scene where Ringo remembers her parents eventually deciding to split up is reenacted via stuffed animals. It's ridiculous, but considering she was a young child at the time it actually works and conveys how traumatic it was for her. And by the same token, most of Ringo's Imagine Spots are so over the top ridiculous that you can't help but laugh, but they also managed to convey just how screwed up the girl is.
For an unexplained reason, Ulquiorra's release has him in a dress and waist-length hair. It stops being cute about 15 seconds later when the Arrancar nearly cuts Ichigo's head off with a pointed pole made of energy.
The English dub of Revolutionary Girl Utena is kinda dorky, and often sounds like a high school drama club performance (Nanami especially), but somehow it seems to work in the show's favor, adding to the general strangeness of the show.
Sankarea is all about a boy who revives a girl as a zombie and their ensuing relationship. It's full of Camp and over-the-top drama, sometimes verging into So Bad, It's Good territory. The later episodes involving the girl's parents (particularly her father) get especially ridiculous. But despise all the silliness, it still manages to be a genuinely funny and entertaining series.
Toriko, the Ou Shoku Bansan/Dinner of Kings. It's a giant Pac-Man Energy Ball that eats everything in its path, and it is AWESOME!
The English dub of Ranma ½ is ridiculously campy. The same is true for the show itself, so it works out.