The Librarian from Discworld, who says only "ook," but can be understood by some characters. There's a Lampshade Hanging in Moving Pictures, where the Bursar tries to explain to the Archchancellor the difference between the "ook" for "yes" and the "ook" for "no," and it becomes apparent he doesn't really know how he understands what the Librarian says.
And eek which seems to mean "ook," but more emphatic.
Discworld actually does it well, with no parrot translations.
Also the Death of Rats only says Squeak, but has no problem getting his opinions across (or if he does, Quoth the Raven can translate).
Given that he hardly talks to anyone but Quoth, Death, Susan, and the rodents he comes for, this might be justified.
Sunny from A Series of Unfortunate Events talks only in babyspeak, which is translated either by her siblings, or by Lemony Snicket's distinctive and verbose definitions. As the series goes on she gradually becomes more intelligible, going from gibberish to slurred words to distinct words to her Crowning Moment of Awesome first full sentence: "I am not a baby."
Razorback of the Whateley Universe can only make weird growling and barking noises. Since he looks mostly like a velociraptor, it's not really surprising he doesn't have a human larynx.
The Blind Side has Ole Miss' Coach O, whose thick Louisiana drawl borders on gibberish to everyone except fellow Louisianan Sean Tuohy.
He plays himself in the movie, and, in this troper's opinion, he isn't that bad, unless he's excited
Baby Kayla from The Giggler Treatment can only say 'a-bah' but everyone understands her anyway, because they love her so much. To the point where Mr. Mack actually mistook 'a-bah' for Kayla's first words... "Mind the poo!" (which were, incidentally, said by Rover the dog. It makes sense in context.)
Hodor from A Song of Ice and Fire says exactly one word: Hodor. In fact, it's not even his name (his real name is Walder). Though Bran occasionally notes that he doesn't know what Hodor's trying to say, it's generally pretty obvious from the context.
David Weber has fun with this — neither the Orions in the Starfire books nor the Rish in In Fury Born have the necessary vocal apparatus to speak English. On the other hand, both races can understand English perfectly well, and vice versa, even if it can be a bit hard on the human ears.
There is one Noodle Character who can speak High Rishathan, but the same person could also "reproduce exactly the sound of a buzzsaw hitting a nail at 8000 RPM"...
Tinker Gnomes in the Dragonlance universe speak extremely fast and have to be reminded constantly to slow down so other races can understand them.
Subverted with Toth from The Malloreon series. Written in the prophecies as "The Silent Man", he is Exactly What It Says on the Tin. He only communicates with vague hand and arm gestures, yet anyone paying attention to him seems to understand him clearly. It takes until the last book, "The Seeress of Kell," for Durnik to finally catch on to what's really happening: his gestures conceal the fact he's actually a telepath.
In Dave Barry Slept Here, Senator Sam Ervin speaks a Deep Southern language "similar to English, only unintelligible."
From Peter Pan and most adaptations, one particularly iconic example of this trope is Tinker Bell's fairy language, which sounds like...well...a tinkling bell.
Gornab's speech in Rumo & His Miraculous Adventures is so messed up, only his chancellor can understated him. OTOH, he consistently manages to speak all words exactly backwards (by syllable, not by sound or letter).
Wuthering Heights has a meta-example in the form of Joseph. It seems the author tried a little too hard to communicate his accent. It leaves you wondering if what is typed is language at all. Luckily there's an entire website dedicated to helping readers understand what he says, one of which can be found here.
Joseph:Yon lad gets war und war!' observed he on re-entering. 'He's left th' gate at t' full swing, and Miss's pony has trodden dahn two rigs o' corn, and plottered through, raight o'er into t' meadow! Hahsomdiver, t' maister 'ull play t' devil to-morn, and he'll do weel. He's patience itsseln wi' sich careless, offald craters—patience itsseln he is! Bud he'll not be soa allus—yah's see, all on ye! Yah mun'n't drive him out of his heead for nowt!'