Goku: (Referring to a giant tree sucking all the energy out of Earth) I think the only course of action left... is to reason with it! Yamcha: Uh, Goku... Goku: Big tree! If you don't leave right now, I'm gonna have to beat you up! Yamcha: Goku, trees can't ta- Offscreen Voice:Why don't you come up here and try it? Yamcha: ... you know what, f*ck it, I don't even care anymore.
An interesting version in one of IDW's Transformers comics, where Shockwave is unable to logically comprehend why the Dynobots (Not Dinobots, not yet anyway) have followed him all the way to Earth for the sake of revenge, and nearly seizes up in the process, which the Dynobots take full advantage of. So Shockwave decides to temporarily shut down his reasoning and see what happens. The resultarescarilyeffective.
By about the halfway point of The Detective and the Diplomat, Sherlock Holmes has been in Ankh-Morpork for about a day with no sleep, encountering wizards, werewolves, trolls, dwarves, and a man transformed into a gold statue. Then he is told that his only witness to the incident that led to the gold statue is a talking dog named Gaspode. After grinding his mental gears between This Shouldn't Be Happening and I Trust My Senses No Matter What for a few seconds, he apparently drop-kicks conventional logic off the Tower of Art and interviews the goddamn dog.
It is a common theme in Concept Road. “Here we go again. I'm with an alien psychic and a gun-spawning magical girl, and people are calling me amazing. Maybe logic has become a thing of the past.”
This comes up fairly often in the Thursday Next books. One example would be the family conspiracy in First Among Sequels, where Thursday's daughter Jenny is often mentioned, but never seen. It turns out that Aornis Hades planted a mindworm, so Thursday thinks she has a second daughter and becomes distraught when she notices she never sees her. Thursday periodically figures it out, only to forget again due to the mindworm's effects. After some failed efforts to convince Thursday once and for all of the truth about Jenny, her husband and other kids give up and play along, acting as if Jenny exists and distracting Thursday when she seems likely to notice (again!) that the girl is missing.
In The Wheel of Time, Mat and two friends travel to the world of The Fair Folk in Towers of Midnight. Mat quickly realizes that logic as it works in the human world doesn't apply there, since things like walking in a straight line only to wind up right back where you started is common. Mat quickly urges the others to stop using human logic and instead rely on his luck to make their way. Since Mat is both Born Lucky and more or less a Reality Warper, it works.
In the classic Star Trek episode "A Piece of the Action," Spock says of the gangster planet, "Facts and logic seem not to apply here." Dr. McCoy is stunned, asking "You mean you'll admit to that?" to which Spock calmly replies, "To deny the facts would be illogical."
The movies show the more gradual and long term version of this trope happening with Spock. As he tells Lieutenant Valeris in The Undiscovered Country "Logic is just the beginning of wisdom, not the end."
In the Star Trek: Voyager episode "Twisted", a strange energy field appears to warp and reshape the interior layout of Voyager. Tuvok attempts to search for crew members using a logical search pattern but is stymied by the alterations. Chakotay on the other hand decides to wander aimlessly, because there seems to be no logic behind what's happened. He actually goes so far as to chide Tuvok for attempting to solve the issue logically. (That said, Tuvok does successfully use logic at then end of the episode: since nothing the crew does seems to have an effect on the field, and it hasn't actually hurt anyone, he says the logical thing to do is simply ride out the effects and hope it doesn't harm them. He's right, and the crew later comes to the conclusion that the energy field was alive and attempting to interact/communicate with them).
Agent Scully of The X-Files remains a staunch skeptic through much of the show, although she does show moments of considering the unlikely. Late in the show's run after David Duchovny left, Scully wound up getting a new partner and becoming the one who was more likely to leap to supernatural and other unlikely explanations.
In The Lonely Island song Jack Sparrow, Michael Bolton goes to collaborate with The Lonely Island boys on your standard hip-hop, going to the club type song. However, Bolton has just come from watching a Pirates of the Caribbean marathon, so when it comes time for Bolton to do the chorus, all he can sing about is Jack Sparrow and Pirates related stuff. At first the guys try to soldier on, but when Bolton keeps bringing the subject back to Pirates and then goes into other movies like Forrest Gump, Erin Brockovich and finally Scarface (1983), they give up trying to deal with Bolton's nuttiness or get the song back on track, and instead sit back with the occasional comment like "You can tell Michael Bolton is a real cinephile."
After all the things that Nick sees over the course of Lollipop Chainsaw, (the least of which is the fact that he is somehow surviving as a disembodied head while watching his girlfriend kill zombies with a chainsaw, which for some reason makes them burst into rainbows) he starts getting used to things that would normally be considered unusual.
Sure. Giant robot elephant. Doesn't even faze me anymore.
The webcomic Final Fantasy VII: The Sevening has a case where Cloud stops trying to make sense of other people's weird behavior and suggestions (like using a leaping dolphin as a means of getting to the top of a 50 foot tall tower) here.
This is an important character development moment for Ardam in ADVENTURERS!.
After the Tom & Jerry movie pulled too many crazy moments, he simply announced "Welcome to the mind fuck!" and let the madness unfold while wigging out and playing "Flagpole Sitta".
In the hybrid webcomic/browser game Demon Thesis a group of four college students are suddenly thrust into fighting Eldritch Abominations on behalf of a manipulative entity speaking in their minds. Since two of the characters are on their school's fencing team, they borrow some swords for a fight. If they give a foil to one of the non-fencers, it'll change into a broadsword, provoking this conversation mid-battle.
Clady: Sam, where'd you get that sword? Sam: Uh... you gave it to me? Clady: I gave you a fencing foil, not a broadsword. Sam: ... Clady: Yeah, never mind. I'm going to stop asking questions like that.
Although there's not a particular moment where it is shown, this happens with Sokka in Avatar: The Last Airbender . Sokka calls water bending magic and laughs at the idea of a flying bison, within a few episodes he gets indifferent to both, (he's since been traveling across the world on said bison and seen his sister and Aang practice water bending) and by the end of the show he just accepts any of the weird things that happen to him, from interacting with spirits, reincarnation, etc.
The My Little Pony: Friendship Is Magic episode "Feeling Pinkie Keen" is all about Twilight Sparkle trying to find a logical reason how Pinkie Pie has the strange ability called "Pinkie Sense". By the end of the episode, Twilight gives up, with An Aesop that not all things need to be completely understood to be considered real or true.
Frank Grimes from The Simpsons memorably went the more dramatic route when he gave up on reason and started imitating the mind-numbingly stupid antics of his coworker, Homer Simpson. Because Frank wasn't Born Lucky like Homer, he dies within minutes.
In the episode of Futurama where he is introduced, Professor Farnsworth's clone Cubert keeps insisting that all of Farnsworths inventions are impossible. By the end of the episode, he gives in to Farnsworth's way of thinking, and manages to figure out how one of his inventions works just in time to save everyone.
In an episode of Family Guy after all her Christmas plans fall apart due to her family antics, Lois finally snaps and goes on a crazy rampage in which she punches Frosty the Snowman's head off.
Happens all the time in The Venture Bros., and is frequently Lampshaded when it does. Most of the main cast doesn't even bother to question truly absurd things anymore, and when a newer and/or more sane character tries to question something bizarre, the other characters will often mention a Continuity Nod or Noodle Incident that makes the current situation seem downright reasonable by comparison. A few specific examples:
In the season one episode "Midlife Chrysalis," Dr. Venture is turned into giant caterpillar. When Hank takes this situation a little too well for Doc's liking, Hank explains that they see weird stuff like this every week.
In the season three two-part finale, "The Family that Slays Together...," there is a final epic battle between an O.S.I. army, an army of Monarch henchmen, and an army of naked Hank and Dean clones led by Sgt. Hatred. Most of those involved don't see this as anything particularly unusual, but it proves to be the straw that breaks the camel's back for Only Sane Man/Badass Bodyguard Brock Sampson. He mentions some of the weird things he's seen over his years as the Venture family bodyguard and quits on the spot.
Discussed by Hank and Dermott in the season five episode "Momma's Boys" when Dermott questions how Rusty could possibly believe that his friend "Teddy," which is actually just a talking teddy bear the boys were using to play a prank on him, is in danger. Hank mentions some of the outlandish things that have actually happened, which make the Teddy situation seem perfectly reasonable by comparison.
Occurs in an episode of The Amazing World of Gumball, "The Curse." Anais, the smart one of her two brothers, tries to dissuade her brother that he has been cursed with bad luck. She attempts to prove it by having him go through several bad luck causing events. She seems to be right, until a cloud forms and strikes Gumball with lightning indoors.
Anais: Guess logic and reason just went out the window.
This comes up again where since "logic and reason went out the window," they should all put their minds to get free amusement park tickets. They were close, but it literally hit the fan.
Darwin: Yay! I wished for snow!
Dadaists, who believe that the things humanity does (such as war) are so nonsensical that art shouldn't make sense either.
Certain unnamed political and philosophical groups who view logic and science as oppressive tools of the powerful.
Presuppositionalists, who view logic as subservient to and reliant upon special revelation.