Insanity Immunity

When a character is already so crazy that none of the traditional Go Mad From the Revelation-type phenomena or Brown Note will affect them in the slightest; they can speed read The King in Yellow and the Tome of Eldritch Lore, get into a Staring Contest with Cthulhu, contemplate the Void without any negative mental effects, and get caught in the Throat of Madness, simply because they can't get any crazier. This can also mean that they can wield/absorb any powers with these effects without any problems, either; powers that would drive a normal man insane simply won't affect them in any negative fashion.

They also cannot be controlled, predicted, or otherwise influenced by others: psychics will recoil in horror from the rotting offal that is their minds, those who can see the future will find their expectations dashed as destiny is royally screwed over, martial artists with Combat Clairvoyance will be no match for their Schrodinger Fu, etc.

This is a Disability Superpower, more specifically a Disability Immunity. See also Too Dumb to Fool and Too Spicy for Yog-Sothoth.

Anime

 * A less extreme example: Osaka in Azumanga Daioh. She's generally less fazed than the others by weirdness, and exhibits less PTSD than poor Chiyo-chan after a ride in the Yukarimobile.
 * Probably because she was thinking about hemorrhoids.
 * Kimblee, in Fullmetal Alchemist, is revealed to have a form of this.

Comics

 * In Lucifer Fenrir manipulates a man with a mental disorder into thinking that he's gone crazy and killed his wife and daughter. This is so he and his companions can "ride his coattails" towards Yggdrasil. By tricking the man into thinking his wife and daughter are at the tree, he gets there without having his mind torn to shreds by the sanity-ripping "thorns" on the path there. The man is more or less immune both by virtue of his insanity and single-minded Heroic Resolve to rescue his family. Don't worry, they were reunited and lived Happily Ever After. (Fenrir, for his part, almost undid all creation.)
 * You must have missed the part where  The guy does in fact lose pieces of himself on the way- some memories, an arm, but his insanity means he's used to surrendering pieces of himself, and thus he can serve as another's chariot.
 * This is about as close as The Joker gets to a superpower. For example, in a Batman/JudgeDredd Crossover, Judge Death tried to possess his body, flowing into his head only to fly out through the other ear, Joker's mind completely incompatible with his mind control powers.
 * J'onn J'onnz was actually able to replicate this effect during Grant Morrison's run on JLA; when he and Superman are trapped in a pocket dimension created by The Joker (which reflects his madness), J'onn uses his shapeshifting abilities to literally change the shape/function of his brain to resemble The Joker's, enabling him and Supes to navigate their way out without being consumed or driven mad.
 * Also, in DC and The Mask crossovers both Joker and Lobo put the titular mask on their faces, and it didn't affect their personalities at all.
 * Not at first, but Joker becomes much more destructive. When Batman points out that this isn't funny (by Joker's standards anyways), Joker realizes that this is true and promptly takes the mask off.
 * Another Elseworld comic had a mass depowering event take place; Joker loses his trademark insanity, leaving him quiet and remorseful over everything he's done.
 * Just to show the different possible takes on it, in another depowering Elseworld he stayed the same.
 * On one team-up with the Scarecrow, Dr. Crane ended their alliance by gassing Joker to see what he was afraid of. Sadly, Joker turned out to be immune to the Scarecrow's gas. So he beat him with a chair.
 * The Joker wasn't necessarily immune. He asked for another hit.
 * In Nextwave Dirk Anger was so crazy that when, his behavior hadn't changed at all (only his diet).
 * In Endless Nights, a companion book to The Sandman, one of the stories is about Delirium becoming lost in her own realm and insane people being the only ones that can enter and leave her realm unscathed.
 * A slight variation - in Dilbert, Wally reacts to news his new bosses can read and control his mind by shrugging and saying if they try to read his mind, they'd kill themselves.
 * Bullseye and Doctor Octopus have both been shown with resistance to mind-control that could either be the result of cybernetics in their heads or sheer insanity.

Literature

 * In Alan Dean Foster's To The Vanishing Point, Burnfingers Begay can deal with the shifting realities matter-of-factly because he's already crazy.
 * The Magic the Gathering novel Final Sacrifice involves a Mind Control Device in the form of a helmet that lets a wizard summon and control another wizard the same way wizards can summon and control ordinary creatures. (Interestingly, this novel was written long before the card Mindslaver was printed.) The druid Greensleeves, having once been insane, finds the effects of the helmet to be similar to the insanity she conquered and is able to ignore its commands - and is also able to access the vast amounts of information stored in it.
 * Billy in Remnants has a mind that clearly does not work normally--he seems to be mildly psychic, has an eidetic memory and rarely interacts with anyone. It turns out he is the only one able to mentally interact with Mother, a Sapient Ship who has Gone Mad From The Isolation, without going crazy. He explains it as his mind being malleable, like rubber, while other people's are like sticks that break if you bend them.

Tabletop RPG

 * Call of Cthulhu. When characters lose Sanity points they can go temporarily insane, which impairs their abilities considerably. Once people lose all of their Sanity points (and become permanently insane), they can no longer go temporarily insane, either due to Sanity loss or certain attacks such as the Mind Blast spell. If this happens to an NPC who is a Cthulhu Mythos worshipper, they can act effectively even though they're completely nuts.
 * Dungeons and Dragons 1st Edition. Psionic attacks could cause various forms of mental disorders in their victims, but insane creatures were immune to psionic attack.
 * There's a "Cloak of Insanity" spell in the Forgotten Realms Sourcebook Menzoberranzan that emulates this effect. It shields the caster's mind from being influenced or read by both magic or psionics, but isn't completely safe in itself.
 * Witches and sorcerers from Pokethulhu, who are immune to anything that requires a sanity save, because they have no sanity at all.
 * Vampires in the Old World of Darkness who drink Changeling blood or otherwise become Enchanted (infused with Fae Glamor, allowing them perceive and otherwise be affected by Chimerical reality) must, due to the inherently Banal nature of vampirism, make a Courage roll in order to avoid succumbing to Bedlam (Glamor induced insanity). The vampires of Clan Malkavian are explicitly said to be immune to this effect as they are already mad.
 * In Dark Heresy, characters ignore fear effects that equal their (insanity points/10)/2: Their minds have simply seen so much sanity-blasting horror already that they've gone insensitive to the little stuff. A character with 80 or more insanity points is literally immune to fear and can stare down a Eldritch Abomination with no ill effects, although at that point that's peanuts compared to the effects the cumulative mental derangements has on that character's mind anyway.
 * One standout example from Exalted is Lilith, a Lunar who spent most of the First Age married to Desus, who kept her in line with regular doses of Mind Rape. She fled into the Wyld after the Usurpation, and actually got better over the centuries. That's how bad her marriage was -- years after years spent in the heart of screaming, primordial madness were effectively therapy.
 * In Ironclaw the Enraged status effect makes the character unable to defend, focus or do mental actions, however it also negates the next mental debuff they would be affected with, (removes Enraged as well). It's more harmful version, Berserk, also does this, but also causes the character to attack the nearest target. Note that for Avarist, Enraged is important for some of their abilities, but helps compensate their low Mind stat.

Video Games

 * In the Touhou game "Imperishable Night", when Cute Witch Marisa Kirisame This is the only time she comes out and says this, however.
 * Implied in Batman Arkham Asylum,, either because of his indomitable will...or his own insanity is strong enough to withstand it.

Web Animation

 * One Halloween episode of Homestar Runner featured a creepy, animate painting that gave everyone who looked at it a paralyzing case of "The Jibblies". Except for Homestar, who was such a ditz that he was completely unaffected. And, according to the Easter Egg, resident Cloudcuckoolander Homsar was also unaffected... and even managed to turn the tables and scare the painting.

Web Original
"Kung-Fu Jesus: He seemed really out of it.
 * Medibot made it through Pokecapn's Let'sPlay of Sonic the Hedgehog 2006 in considerably less psychological distress than the other three who made it to the end (there was a fifth guy, John Condit, but he wisely bailed partway through). This was partly because he left to take a nap partway through, but mostly because:

pokecapn: As he always does?"

Western Animation

 * Notably subverted in Justice League: the Joker was using Ace, a girl with telepathic powers, to drive crazy as many people as possible. It turned out he only claimed and/or thought he was immune, until Ace focused her attention specifically on him.
 * It revealed in Batman Beyond that he recovers from this, so The Joker might not have been too far off.
 * Completely inverted in Batman the Brave And The Bold, when Joker gains the powers of Batmite. After torturing Batman the entire episode, Batman tricks him into trying to drive Batman insane by entering his mind. Turns out
 * Or maybe  Afterall, He's the goddamn Batman!