"Do you – do you think I want to – do you think I give a – I DON'T CARE WHAT YOU'VE GOT TO SAY!"
Glen Cook plays with this in his Garrett, P.I. novels, having characters who are furious and/or recovering from a blow to the head speak in Angrish ... or seem to, as in one case it's revealed that the gibberish-speaker is actually cursing fluently in Elvish.
Stephen King's Cell involves a Hate Plague "Pulse" afflicting people using their cell phones, sending them into berserk rages while gibbering Angrish.
The Dark Tower series gives us, The deranged shrieking of the Crimson King.
Belisarius is reduced to this in Destiny's Shield, when Antonina refuses to take along guards on her trip to Egypt.
In Malevil, Momo is reduced to this when their wheat harvest comes under attack. What's significant is that Momo is barelycapable of speech in the first place.
In The Eye of Argon, the evil prince Agaphim is reduced to speaking "muffled sibilant gibberish" when Grignr insults him. The "faintly perceptible ceremonial jibberish" (sic) spoken by a cultist earlier in the story may also count.
Zilpha Keatley Snyder has David Stanley do this in The Headless Cupid. He is slightly shy and reticent because he's been teased for stuttering when he's angry. His half-sister finally infuriates him to the point that he doesn't care.
In Fear and Loathing on the Campaign Trail '72, Hunter S. Thompson imagines George Meany's reaction to George McGovern winning the Democratic nomination, and the imagined bout of angrish lasted eight minutes:
"He raged incoherently at the tube for eight minutes without drawing a breath; then, suddenly, his face turned beet red and his head swelled up to twice its normal size. Seconds later—while his henchmen looked on in mute horror—Meany swallowed his tongue, rolled out the door like a log, and crawled through a plate glass window."
In John M Ford's Star Trek novel How Much for Just the Planet? a series of coincidences leads a Klingon officer to believe that he has walked in on Captain Kirk cheating with his wife. Kirk is unable to determine whether his subsequent utterance is actual Klingon speech or just Angrish, since they normally sound very similar anyway (and he's rather distracted with his efforts to flee for his life).
In Jack Blank, late in the book, Jack goes on an angry tirade using a "great deal of words not commonly found in [Jazen's] vocabulary after finding out that his birth certificate file is not infected and Jonas Smart was lying to him about it
In "Band-Aid", a story in Andy Griffiths' Just Crazy!, when Andy tries to pull a bandage off from under his eye, he describes the pain in three pages' worth of Angrish.
In 1634: The Baltic War, history teacher Melissa Mailey learns that Harry Lefferts and his gang set fire to the Globe Theatre to cover their escape from the Tower of London. Which they also blew up. Oh, and they blew up London Bridge, too... or part of it, anyway. She does manage the occasional coherent sentence, but her reaction is largely described to be shrieking and gobbling. From Melissa Mailey, this reaction is entirely expected.
Harry Dresden has a quite justifiable bout of angrish after learning in Cold Days that, on top of everything else he has to deal with already, the island Demonreach is getting ready to explode and take out a fair chunk of the Midwest.