This is implied rather heavily to be part of the reason for the Japanese Super Young Team's awkward sounding names in The DCU, the other being simply awkward translation. Big Atomic Lantern Boy's name is almost certainly entirely in English, though.
In American Born Chinese, a new immigrant student from Taiwan first appears with a shirt that reads "Happy Robot".
In Kyon: Big Damn Hero, Kyon was forced to cosplay in a Victorian suit. He accepts tea from Yuki, who was in an Elegant Gothic Lolita costume, with gratuitous, but grammatically correct, English. Mikuru nearly squealed in delight.
In The Vampire Countess there is an Englishman among the conspirators who is described as being Welsh. At one point he says something in what Paul Féval calls a Cockney Accent. What Feval wrote was "Let us knock down the rascal". In his English translation Brian Stableford chose to simply replace this all together as "Flatten the bleeder".
The Japanese Harry Potter books, in addition to featuring the English title as well as Japanese on the cover, have English romaji beside the kanji for their publishing company - and it's written "Say-zan-sha (Seizansha)."
Lampshaded throughout Chris Jericho's first book, A Lion's Tale, while recounting his times in Japan. He often referred to it as "English just good enough to make no sense."
In the Japanese translations of the Warrior Cats series, the names of the characters, which are usually combinations of nouns, verbs and adjectives, are left in English.
No less a philosopher than Friedrich Nietzsche was known to drop English (as well as French, Latin, and Greek) into his otherwise-German works. He usually used this when quoting from an English work, but sometimes used English words alone to make a point, to screw with the reader, or just because he felt like it. Since he spoke English, he knew exactly what was being said (so no funniness from misplaced words) but it makes reading Nietzsche interesting for English-speakers: if you're reading it in English translation, the footnotes that say "this bit was originally in English" are often kind of amusing, and if you can speak German and are reading it in the original, it's rather shocking to see the English in a sea of German.
A French translation of a Peanuts comic strip once had Linus Van Pelt, after speaking in perfect French, refer to his blanket-hating, caffeine-addicted grandmother as "Granny." Pretty jolting, especially since the French almost certainly have a pet name of their own for their grandmothers.
ThisU.S. Acres strip has Orson reciving a phone call from China after sneezing. Guess what's wrong with the message.
Professional Wrestling in Japan provides a pleasing real-life example of this trope. Since the conventions of pro. wrestling were adopted wholesale for the Japanese version of the sport ("puroresu" — itself an example of Gratuitous English), all the names of the moves are the English ones (except those invented in Japan, like the enziguri), which the announcers faithfully reproduce in commentary, even when they sound ridiculous. Examples include "DIIIIVING BOOOOOODY AAAAAAAAAAAAATAAAAAAAAACK!!!!!!!" and the famous "LAAAAAAARIIIIIIAAAAAT-OOOOOOOOOOOOOOOOOOOOH!!!!"
and the now-ubiquitous "SHIIIIIIIIIIIIIIIIING WIIIIZAAAAAAAAAAAAAARRRRRRRRRRRRRRRRRRRDOOOOOOOOOOOOOOOOOOOOOOOOH!"
Dragon Gate theme song names frequently run along this trope, with names like WILD DRANK HUSKEY (Don Fujii) and KICK START THE ELEPHANT (Yasushi Kanda). Yes, they spell the song names in ALLCAPS.
A lot of classic Zenjo tag teams name fall into this. Some, like the Dynamite Girls, sound normal enough. Others, like the Queen Angels or Beauty Pair, straddle the line. And then you have teams with names like Marine Wolves and Dream Orca...
The "Alabama Song" (which, of course, was Covered Up by The Doors) and "Benares Song" in Aufstieg und Fall der Stadt Mahagonny (The Rise and Fall of the City of Mahagonny). Bertolt Brecht's use of English is rather awkward in the latter: "There is no boy with whom to shake hands. Where is the telephone? Is here no telephone?" In the former, the Shotacon implications of the line "show me the way to the next little boy" were doubtless unintentional.
Madama Butterfly throws off a few Gratuitous English phrases, most notably Pinkerton's toast to "America for ever!"
Played for Laughs with Dr. Bent van Helsingřr, a recurring character in the long running yearly Danish show Crazy Christmas Cabaret. He constantly tries to translate untranslatable Danish expressions and proverbs to English, causing him to either sprout complete nonsenses or unintentional sexual innuendo.
France Five, despite being a French series, shares the love of Gratuitous English of the Sentai shows they parody.
Menelaos from Greek Ninja speaks English by making direct translations from Greek, which of course results in terrible mistakes and people not understanding him. Eleonora often takes the role of the translator between him and the rest of the world, having full knowledge of both languages.
This song is kind of stupid It doesn't make sense The English is all fucked up That's okay [we do it all the time!] [Hey hey, let's go] fighting The important thing is to [protect my balls] I'm baaaad, [so let's fighting] [Let's fighting love — let's fighting love!]
That phrase "Let's fighting" is an example of what is, tragically, a very common Engrish construction in Japan. The bowling episode of Mega Man NT Warrior has a bunch of characters repeat the catchphrase "Let's bowling!" — making it perhaps the only one that's more painful to watch subbed than dubbed, ShoPro and all.
Seacht has quite a few English words mixed in with the Irish dialogue; this is particularly surreal as the series is set in Belfast, and one would think that this means the characters are actually speaking English.
Icy in Winx Club, or at least the French dub, has a couple of attacks with English names. The original Italian also has plenty, considering the heroine is named Bloom, there's a character named Icy, and "Winx" is a pun on wings. Whether this is due to it being influenced by anime, or due to the creator's wife being Singaporean, is up for debate.
English language cartoons end up with this trope when translated for a Japanese audience as many bits of the original dialogue and song lyrics (if there are songs) are retained as is for various reasons.
This is more prevalent in The Boondocks' Japanese dub, due of the use of some words (like nigger) whose Japanese equivalents are forbidden to use in Japanese media, so the translators used the original words untranslated from English instead.
Same case in Japan with South Park, but less exaggerated.
In Tom And Jerry during the 'Mouseketeer' shorts which took place in France, Jerry was accompanied by a little gray French-speaking mouse named Tuffy, who occasionally threw English phrases into his speech. In one short, when giving a long-winded explanation in French as to why he ran from Tom, he ends it with "and besides that, I'm chicken!"
In a non-Japanese example, Metalocalypse gives us Swedish Skwisgaar and Norwegian Toki, who both suffer from the same ignorance of the English language. They both have atrocious problems with putting excessive plurals at the end of words (whether or not they are nouns in the first place), frequently use "am" for any form of the word "be", and have a bad grasp on vocabulary in general.
Oh Toki, its adorables, you really wants to takes more solos, but I am the lead guitarist, you know, why? Because I ams, hows do you says, way more gooders than you.
In all non-English foreign versions of Dora the Explorer except for the Turkish, Serbian and Irish versions, the characters speak in Gratuitous English.