Cousin Itt in The Addams Family. Other members of the family have no problem understanding him, but other characters have trouble.
The premise of the British Game ShowThe Adventure Game is that the contestants have travelled to Arg, a planet whose Sufficiently Advanced Alien inhabitants, the Argonds, usually resemble dragons but turn into humans for the contestants' visits. In their Argond form, their speech was incomprehensible babbling (that was only occasionally subtitled); the Argond monarch, the Rangdo, spent Series 2 and 3 turning into an aspidistra instead of a human, and Series 4 turning into a teapot, making his speech just as incomprehensible as it was in his usual Argond form. Many of the contestants also found the backwards-speaking Ron Gad an example of this trope... unless they realised he was speaking English, but backwards, and they had to do likewise to converse with him.
Bear in the Big Blue House has the lemur Treelo who can be hard to understand even when speaking normally and often descends into a sort of babble that closed-captioning refers to as "Treelo-ese." Johnny and the Sprites, which was made by many of the same people, had a very similar character named Root.
Henrietta Pussycat on The Childrens Corner. The Children's Corner was an early show with Fred Rogers on a local station in Pittsburgh, before both the Canadian Broadcasting Corporation's Mister Rogers and PBS's Mister Rogers' Neighborhood. Many of the make-believe characters from Mister Rogers' Neighborhood had their start on The Children's Corner, including Henrietta Pussycat. Henrietta could only say the words "beautiful," "telephone," and "Mister Rogers," replacing all other words with "Meow," placing her in the ranks of The Unintelligible. On the later shows, although she still replaced many words with "Meow," she spoke enough English words so that the audience could understand her meaning, and thus she no longer qualified as The Unintelligible. Although Henrietta referred to her owner, Mister Rogers, he was never seen on screen, making him The Unseen in that show.
The Diagnosans in Farscape speak in a high-pitched trilling language far too complex for Translator Microbes, and as most of them only know a few words of comprehensible speech, they are forced to rely on an interpreter.
The Pilots also have this problem, though most of them have learned to speak simply enough to be understood: however, the Pilot of Rovhu in "Eat Me" appeared to have reverted mostly to his own language, mixed with frequent terrified whimpering.
Father Fay of Father Ted communicates entirely in monkey-like grunts, but is apparently still capable of giving stirring, emotional speeches.
A Song For Europe features a TV presenter who rambles and mumbles in a very thick Irish accent, understood by only his producer (who also happens to be his gay lover). Later averted in the same episode when he speaks in a very clear and understandable voice on stage. However, afterwards backstage he reverts back to being completely non-understandable.
Australian comedy show Full Frontal featured the character of Milo Kerrigan, who spoke in unintelligible gibberish (supposedly because of damage to his vocal cords, as he was a retired boxing champion) but other characters always understood him perfectly.
A guest on the German talk show Hans Meiser. A snippet subsequently became a Running Gag on the comedic show TV total. A decade later, the TVT crew said they still hadn't figured out what the guy was saying.
In It Ain't Half Hot Mum, Rumzan would always speak in Urdu without translation, ending with a few words of English such as "... one of those days!"
The window cleaner on The IT Crowd who talked Roy into keeping his bike in Roy's flat and was very persuasive, although Roy isn't sure about that.
The IT representative that Jen and later the police call for help with her laptop. His speech sounds like a weird mix of Swedish and French accents.
Jen once pretends that she can speak Italian in order to wow her bosses by acting as interpreter for a perspective business partner. Initially she says some gibberish that turns out to translate to "Everyone likes cat smell" and "Spiders". She then starts using translation software on her laptop, but shit hits the fan when it comes time for a face to face meeting and they won't let her bring the laptop into the room. She keeps up the rouse by mumbling gibberish (including "Super Mario") causing the Italian speaker to get madder and madder at his inability to understand her. "I'm sorry, are you the same person I spoke with on the phone? I'm having a hard time understanding your accent."
The League of Gentlemen had Pam Doove, an actress who becomes The Unintelligible whenever it's time for her to speak in character. She auditions for an orange juice commercial and gets the part despite her inability to pronounce a simple line clearly, and the subsequent career boost lands her the part of an equally Unintelligible Nazi in a stage production of The Diary of Anne Frank.
Inverted in Little Britain: Meera (an Indian) speaks perfectly clear English, but the racist Marjorie acts like she doesn't understand her one bit. In one of the specials, Marjorie went on an exchange to America where she pulled the same stunt with a Latina woman in her slimming class.
Lanny from Lizzie McGuire does not speak at all, nor does he use sign language, but Matt has no problem at all interpreting his nods and smiles, to the confusion of Matt's parents. A smile or a tilt of the head from Lanny can be translate into detailed plans and information. In at least one instance, Matt and Lanny hold a telephone conversation without apparent difficulty.
Melina seems to be able to understand him as well. Maybe it's just something about the age?
Not necessarily. A random repairman, who was appearing for just the episode, seemed to understand him perfectly while the McGuires were left in confusion.
When Lizzie swaps bodies with Matt, she can understand Lanny, much to her amazement.
Jin-Soo Kwon from Lost, a Korean, is the only character who can't speak English at the beginning of the series. His wife Sun, however, speaks English proficiently. Jin's English improves over time as he picks things up and Sun teaches him. Ironically, Kwan's portrayer Daniel Dae Kim was raised in Los Angeles and grew up in a mostly-English-speaking household. In a reversal of their roles, his costar Yunjin Kim (who was born in Korea and lived there until the age of 10) teaches him all his Korean parts on the show.
Also, by the end of the first season, Michael, who is working on a raft with Jin can understand his Korean, although the other characters can't. One character, Sawyer, even refers to Jin as "Chewie".
In a reversal, one episode shows the English-speaking characters from Jin's point of view, which sounds like unintelligible garble.
Kelly from Misfits has such an unusual accent that the other characters can't understand her at first, and some fans claim that they still can't. Turns out it's just a really strong Derby accent (you don't tend to hear them on television, or...anywhere outside of Derbyshire really, so most people aren't familiar with it). Those who meet the actress are often shocked to discover that yes, that actually is her real accent - and if anything, she tones it down slightly for the show!
Kermit: (over the Chef talking) I know, I know, I know, I know, I know! (Chef exits) I don't know.
A recurring gag in the series and other Muppet material is to give them songs — separately or a duet, or even a trio with Animal, who has a very limited vocabulary and tends toward Hulk Speak. Thus, you end up with a song that's entirely or largely unintelligible — The show's rendition of Danny Boy, or the web short Habanera are good examples of the result when you get all three together. Their rendition of "Carol of the Bells" is another good example.
One sketch had the Swedish Chef readily admitting that he speaks Mock Swedish and not actual Swedish (to Jean Stapleton, who is apparently fluent in Mock Swedish) and that his real mother tongue is Mock Japanese.
Camila the chicken and Gonzo's girlfriend is the only of the regular muppets that is not a Talking Animal and indeed only makes chicken sounds, yet she's part of all the musical numbers. Gonzo has no problem understanding her tho.
Creepy Child Boris in My Spy Family is shown rarely, speaks rarely and when he speaks he does it only in Russian and without any subtitles, making him completely incomprehensible to the audience.
Ozzy Osbourne parodies his own manner of speech in one commercial spot. His mumble is so bad that he can't even order coffee without sending a text message to the barista. When he is having a therapy session afterwards, his therapist asks how this makes him feel. He mumbles, then sends a text. What did he say? - 'Like I want to shave my bollucks'!
He plays it up again in a later commercial where he laments the numerous failed attempts at having servants over the years. He explains one servant who he claims must have been deaf, then a flashback shows that the servant was not deaf, but rather unable to understand Ozzy's speech.
PBS Kids Sprout's mascot Chica the (puppet) chicken. Her voice is provided by a kazoo and she never appears without someone who can handle Repeating so the Audience Can Hear duties. This hasn't stopped her from getting her own series (The Chica Show), in which everyone else understands her perfectly. It's implied that she's a child, and since her parents Mr. and Mrs. C speak perfect English she would likely become intelligible if she were to mature. It's come to the point that not everything she says on her starring show and the original Sunny Side Up Show block is "translated" outright by others.
Eugene Root in Preacher (2016), AKA Arseface although he's not called that in the show to date. His disfigurement isn't as pronounced as it is in the comic, but his jaw and mouth are still destroyed; his dialogue is subtitled, but Jesse can understand him just fine.
And then there's Buckwheat. In the commercial for his album "Buh-Weet Sings", Buckwheat sings a song (actually "Bette Davis Eyes" by Kim Carnes) where, instead of the title, a bunch of question marks appear on the screen.
Sonny on The Sean Cullen Show, who shouted out nonsensical syllables. In one episode, a speech therapist tried to help him, and mistook his unintelligible shouts for correctly pronouncing the word 'Marne', a valley in France.
The low talking girlfriend of Cosmo Kramer on Seinfeld. Only Kramer seemed to understand what she said. (OK, so she once says something intelligible: "YOU BASTARD!!")
Kramer is briefly this after being kicked by Crazy Joe Davola. He starts speaking on the telephone in God-knows-which-language.
Also, the Tamarians from the Next Generation episode "Darmok". Speaking in metaphor will do that.
It appears that the universal translator can decipher the Breen's speech, since other characters show no problems with understanding them. The audience, on the other hand...
"Daddy" in Stella. Everyone in Pontyberry can understand him perfectly, but outsiders can't.
Don Vito on Viva La Bam, so much so that the subtitles are equally incomprehensible.
In Whose Line Is It Anyway?, the Sound Effects game involves Ryan Stiles mouthing all the sounds as Colin Mochrie wordlessly improvises his scene — any dialogue involved comes out as an indistinct murmur. Most of the time.