The strength of a madman is proverbial, and with good reason. Madness allows people to brush aside the instinctive and conscientious inhibitions that stop the rest of us using our full potential, and perform feats of great strength. Unfortunately, this is also very likely to throw the person right into the prison or the cemetery. Obviously, the same principle will work for supernatural powers. The mad can break through all those barriers meant to keep humans safe, and access their true potential, which would be nice if they weren't stark raving bonkers.
Particularly common forms of this include:
MadGenius - Somehow no longer seeing the world as others has unlocked a brilliance that is far beyond what a sane mind can achieve. Until you apply it to those few select areas at least.
It's got to be especially maddening if they realize that they don't exist, but are figments of someone else's imagination. It's a little better if they realize that they are nothing more than a representation inside a massive computer.
Kaori Sakiyama (you must say her full name, it's a rule) in Air Master. Little training, yet is a formidable martial artist against anyone except the main character simply due to her insane rage against her lot in life. She has even unlocked special attacks simply because she's that ticked off.
In Fate/stay night, Berserker class Servants have a Mad Enhancement ability that increases all of their stats by one rank at the cost of their sanity while in use.
The madness of the Demon God Ashura not only warped reality around him slightly, but actually was infectious, forcing those near him to see threatening hallucinations and more importantly, if left to run rampant, the madness would blanket the world, driving everyone insane.
The black blood is an even better example. Since Chrona and Maka gain a great deal of power from being insane under its influence.
Later on in the manga, (the anime didn't get that far) Insanity is a full-blown phlebotinum. It is researched and used for attacks, creating "clowns" from it for additional power boost or protection, etc. The main researcher is Medusa, the creator of above-mentioned black blood. She herself is notably sane and chastised by rival factions for creating "fake clowns". Noah's fraction, notably Justin, are more about using natural White Clown.
Later on, Black*Star and Kid both utilize insanity as well, having earned the favor of the Eldritch Abomination from the Book of Ebon. They were both already somewhat crazy anyway, so it's a fairly logical step to just go the whole way. Speaking of the Book of Ebon, said Eldritch Abomination, the Black Mass, actually represents Madness born of Power.
All characters make use of the phlebotinum, but Kid is a slightly different instance, seeing as - if one believes the Salvage arc - he's a clone/Soul Jar of a Eldritch Abomination (The Grim Reaper himself), who in this universe are (supposedly) incarnations of madness - Madness of Law in his case, foreshadowed by his Super OCD. Also demonstrated that these beings are susceptible to one another, as Kid reacts badly to both Asura (fear) and the Great Old One (power) inside the book.
In the end, though, madness is revealed to be a Superpowered Evil Side, that incarnations of concepts like him can use to vastly increase their power and become virtually indestructible, at the small cost of destroying minds of everyone else by just existing. While Ashura, drives people mad with fear, Kid after assuming his father's mantle would have turned them into his mindless puppets of his law, had he succumbed to temptation of using madness as well.
Stein also makes use of the madness to gain advantage in battle, in his case the madness was always there (increasing since around half of the story), it was just being repressed; in the Battle of the Moon, he releases all his madness in the beginning and then regains control latter in the battle.
Espers in A Certain Magical Index gain their powers based on their own warped inner view of the world being so powerful that it actually affects the world around them. When you look at the level fives, it becomes only too clear what this means for a person: At least three of them have committed murder, one is a paranoid mind control user, one's identity is unknown because they never appear anywhere and the last one is convinced that he's a superhero. Only Mikoto Misaka is actually well balanced and normal, making her the poster girl for the level fives by default despite her own personality flaws.
Unlike other homicidal maniacs in Black Lagoon, Roberta becomes far more powerful and dangerous when she goes off the deep end, with complicated strategies and maneuvers that lunatics like Hansel and Gretel distinctly lacked.
Deadpool is an interesting example of both With Great Power Comes Great Insanity AND this. His lunacy is due to having been imbued with superhuman regeneration while suffering from terminal cancer, which means that his brain is constantly rotting away and regenerating back. However, this same insanity grants him several perks that increase his power even further.
Peter Milligan's Shade, the Changing Man is the prime example of the Reality Warper ("forge what you need on the smithy of your soul".) He began merely poetic, and therefore only insane to his native culture, so he was able to survive being flung through the Area of Madness relatively insane. With time on Earth, he got much madder.
Hulk's rage and anger is shown to be so powerful that not only does it cause Unstoppable Rage but also gives him immunity to mind control and other telepathic attacks, often showing him simply shrugging it off. Also Banner's fear of his father coming back from beyond the grave has given him the power to see ghosts, and astral forms.
The fact the Hulk exists at all may be an example of this. When Banner first got hit by gamma radiation, his various personalities manifested themselves as the Hulks.
Gamma radiation based powers seem to work like this in general. Gamma radiation brings out everything people suppress (assuming it doesn't just kill them like real radiation poisoning). Depending on what they are suppressing, the resulting transformations can be...unpleasant.
Max's mind is so screwed up he's immune to all forms of mind control and hypnosis, has various psychic powers (which mostly served as Deus ex Machina or one time gags until The Devil's Playhouse) and awareness of the fourth wall. Granted this awareness is shaky, so is his awareness of his own world.
Dr. Will Magnus, creator of the Metal Men, does crazy things when he's not on his meds. Specifically, he has bipolar disorder. During his creative highs he's capable of building a superpowerful and destructive robot made of plutonium. During his soul-crushing depressive lows he feels horrible enough to seriously consider activating it.
In the Harry Potter fanfic Flying Without A Broom, the Death Eaters spike Harry's drink with recreational wizarding drugs, the effects of which are indistinguishable from Harry temporarily losing his mind. The Death Eaters thought it would render Harry helpless, but instead it lets him tap into vast magical powers, powers which he uses to keep himself amused.
In the Bleach fanfic Uninvited Guests, after the first "arc", Hitsugaya goes insane and decides that the best way to solve his problems is to charge headfirst, alone and with no sort of plan whatsoever into Las Noches and beat the crap out of Aizen. It works.
In the movie version of Silent Hill, Alessa's powers are born of overwhelming hatred and despair, which ultimately adds up to madness. Plus, have you SEEN the Abandoned Hospital slash Black Bug Room she calls home!?
Apparently this is the reason "Howling Mad" Murdock from The A-Team is recruited in the first place. The movie version of him can make various airborne vehicles perform in ways that defy all laws of physics and decency.
Murdock appears to be able to fake any foreign accent required (but often just for fun) and speaks Swahili at a customs desk, despite showing no previous knowledge of foreign languages to his teammates or to the audience.
He also appears to be immune to electroshock therapy, even enjoying it while the frustrated doctors keep cranking up the voltage to no avail. It takes nothing less than a bullet to the head to make him feel sane for a while.
Gallian in In the Name of the King: A Dungeon Siege Tale is one of the last magi. He figures out how to be independent from servitude (a magus must serve a king) by naming himself king of the mindless Krugg. He then forges them into a fearless horde that he uses to try to conquer the kingdom of Ehb. He also gains more power by sleeping with Muriella, the daughter of Merick, the king's magus. When confronted by Merick, they converse while dueling with magical swords suspended in mid-air. Merick finally realizes that Gallian has truly gone insane with power. Gallian doesn't deny and shows him exactly what madness can do, even though Merick is the older and more experienced magus.
Gallian: You have no idea... how powerful madness can be.
Insanity seems to be the primary source of physical strength for the antagonists of the Scream series. In the second installment, even a middle-aged woman with no sign of physical training is able to overpower and kill young men who should be in their prime.
In the Star Trek novel, Q-Squared, Trelane is more powerful than his fellow Q because of this.
In another series of novels, the Q-like entity 0 has gone insane after being locked out of the can for millennia, and thus has no limits on what he can do. Q specifically invokes this trope by saying, "0's insanity gives him an insane amount of power."
Big Fido, the Canine Supremacist in Men at Arms. He's a small poodle, but his madness gives him the power to rip a man to shreds, and cow all the other dogs through sheer force of personality.
Agness Nitt's split personality allows her to resist mind control. When one personality is dominated, the other takes over.
Carcer and Jonathon(?) Teatime as well, both psychotic, but it took Vimes and Susan respectively to bring them down.
Carcer was, as it were, worse, because he wasn't psychotic. He'd passed through insanity and circled around to sanity from behind. The thing that made him dangerous was that he had realized that all of society's rules that keep it ticking over nicely only apply to you if you let them. And he didn't want to let them.
Vimes fits this himself on those occasions he lets 'the beast' off its leash.
In the Mistborn trilogy, by Brandon Sanderson, allomancers come into their powers after Snapping (something like a breakdown after a horrific event) - the most notable being Kelsier, who came into his powers after spending an unidentified amount of time working in a really nasty mine, followed by watching his wife being beaten to death before his eyes.
Lampshaded at least as early as Dracula where Renfield mentions having heard of the strength of madmen, and reasons that as he's mad he must have it. He successfully uses this strength to knock Dracula out of his mist form and, for a moment, get the upper hand in the fight.
From the Nightside, there is the one known as Madman, who was driven mad by perceiving the truth underlying reality? He's a dangerously powerful Reality Warper whose hallucinations manifest around him. Most notable is that he's accompanied by his own personal soundtrack. The other one is Jessica Sorrow the Unbeliever, who thinks that everything is just her hallucination and everyone is just another voice in her head. Just focusing her unbelief, she can unmake anything.
In This Alien Shore by C.S. Friedman, "Outpilot's Syndrome" is paranoia dialed Up to Eleven, and those suffering from it require massive amounts of medication to do more than lie around whimpering. However, Hyperspace Is a Scary Place, so scary that they're merely Properly Paranoid when in it, so they're the only people who can pilot spaceships through it without getting eaten by Eldritch Abominations.
In On Stranger Tides, hero Jack Shandy suffers a blow to the head just before confronting an evil sorcerer, who attempts to trap him in illusions. The illusions keep breaking down as a result of Jack's disorientation, causing the villain to exclaim: "What's wrong with your mind? It's like a stripped screw!"
The Dresden Files: Molly Carpenter by Ghost Story, who displays how completely terrifying a master of illusion can be and how fast a newbie can advance if motivated strong enough. Insanely strong.
In the same story, when told that only truly insane or insanely determined ghosts can make themselves solid, Harry runs over his past exploits (Zombie T-Rex springs to mind) and concludes that he must be one or the other. Which one isn't clear — but it works.
In David Brin's Uplift series there is a species of client-alien called the Episiarchs, which have been genetically engineered to be so insane that they have the ability to alter reality by refusing to accept it.
As well, Dalek Caan, who managed emergency temporal shift into the Time War, which was supposed to be impenetrable. The process drove him mad, but in the process, he saw everything.
The reason the Reavers from Firefly are so dangerous is more because of them being insanely furious all the freaking time more than anything else.
Helen Magnus in Sanctuary engineers a Batman Gambit to catch a bad guy that involves pretending to kill a friend and putting a critter in her brain that will make her seem insane to throw off the psychic powers of those meant to determine if she committed the murder since one of them was aforementioned bad guy.
The only real way to describe Boogeyman's quirks, such as his ability to evade and bypass all security measures and the difficulty in getting him to sell anything up to and included shards of glass buried in his skin.
Warhammer 40,000: "In the darkness, a blind man is the best guide; in an age of madness, look to the madman to lead the way."
To be precise, psykers have incredible powers, but as a general rule are batshit insane, to the point that burning witch-erm, psykers, at the stake is in fact a perfectly reasonable and wise precaution. In places where laws of physics go to hell (such as a planet engulfed in a warp storm) sufficiently mad people can become spontaneous reality warpers. And each and every psyker across the galaxy is at constant threat of CREATING one of these places. Every second of the day. Along with breaking the barriers that hold of the entirely omnicidal Powers-that-Be in the package. Along with a horde of daemons. Forget reasonable, it's only because the Imperium is so stringent about shepherding psykers up into the Black Ships that the Imperium's 'only' teetering on the brink of utter annihilation from every named force in the universe. Darker and Edgier indeed.
The various tabletop RPGs have several mechanics to track your progress: aside from experience, there's also "insanity" and "corruption". In most of the games these are bad things to build up... Except in Black Crusade, where more corruption gives you more Gifts of the Gods, but there's no insanity track any more. The assumption is that you start the game at the maximum of 100 points of Insanity, and it's only by being that off the wall that you even CONSIDER tapping into the kinds of power granted by the Chaos Gods.
In Unknown Armies, all mages works on this principle. Ritual magic can be done by the sane, for rather wide definitions of sane. The faster and more powerful adept magic, however, can only be powered or cast by individuals who are crazily obsessed with money, books, historic places, alcohol, or danger to the point of that obsession overwriting local reality. A typical bibliomancer, for example, would be so obsessed over books that he or she would cajole or even kill for a given text, and take an assault on the Library as a personal assault. Mechanically, becoming an adept after character creation usually requires five failed Stress checks in a single column and the resulting madness. GMs are instructed to punish adepts who don't let their character's insanity shine through with drops in power or odd spontaneous magical effects.
Old World of Darkness: the Marauders from Mage: The Ascension. Something went wrong during their Awakenings, resulting in them seeing reality through a skewed lens. They are extremely resistant to the effects of Paradox, which can sometimes slide off them and affect any other mages in the area. If they get crazy enough, they're eventually shunted off into their own pocket dimension in the Umbra.
The Malkavians in Vampire: The Masquerade are driven insane the second they are turned, and this very insanity powers their unique ability to cause delusions and hallucinations in others, as well as see 'patterns' in reality that their saner fellows cannot. In the computer game, Bloodlines, Malkavians also get different dialogue options, many of which hint at or warn about future events... albeit in such a confusing manner that you need to finish the game through a sane character first to get all the references. Malkavians are also able to ignore the existence of certain objects rendering them useless against that character, e.g. the machete that did not repeatedly hack into their necks. In games where most of the PCs are Malkavians, it's more a case of the NPCs hallucinating the object...
The werewolves of Werewolf: The Apocalypse are angry. Angry enough for some to cannibalize their own species or molest allies to death if they don't keep a lid on it. Angry enough to fuel regeneration or shapeshifting through rage alone.
In Genius The Transgression, Geniuses wouldn't exist if their mind wasn't influenced by a mental breakdown fueled by discovery. This often leaves them quite mad. Consequentially, their Wonders, while not quite as blatantly as the Mages' magic (for example, the sourcebook says that simply putting Mania into a wooden bird won't make it fly just because it's "enchanted"), can bend or break several important laws of physics. And they are quite explicitly not science as they're non-repeatable phenomena, which leads to...
Changeling: The Lost has "Beautiful Madness" as a theme of the game, and the ongoing challenge of a Changeling's life is to strike a balance between the stolid, dependable order of the mundane world and the wonderful, awful power of the Wyrd. More powerful Changelings are almost guaranteed to be at least a bit crazy - but they can also force the world to play along with their ideas of how it should work. Even the weaker ones can, for example, weave moonlight and daydreams into an invisibility cloak, because it doesn't occur to them that you shouldn't be able to thread a needle with either of those things.
The Eldritch Order of the Lost Pantheonnote (In the Lords of Summer sourcebook) deserves special mention. Every one of them is very powerful, somewhat insane, and believes to their bones that they are the embodiment of an archetype that determines how they make sense of the world. It works: their madness makes them more perceptive and longer-lived, and gain power from mortal worship.
The True Fae. Without exception, they have no understanding of objective reality, and define their existence by narrative. While this necessarily includes an Achilles' Heel, it can also make them otherwise invulnerable as long as they stick to their story.
And, of course, regular Changelings can access some of this power by raising their own Wyrd high enough, going completely insane, and metamorphosing into True Fae themselves.
In the world of Exalted, when The Fair Folk use up all their Virtue Channels for one of their Virtues, they enter bedlam, a state of absolute madness. Some Fair Folk charms are made specifically to benefit from this state when it happens.
There are also a number of Charms for the various types of Exalt that use the madness of the Great Curse to their advantage. For example, Stubborn Boar Defence allows a Solar in Limit Break to ignore unnatural mental influence.
It isn't strictly limited to Charms; not fighting Limit Break (which usually translates directly "flow with the crazy") yields in-game advantages, such as a maxed Willpower pool.
Failing a Madness save in a Ravenloft game is usually bad news, but a couple of possible outcomes do provide fringe benefits, e.g. a bonus to Fear saves because you don't think the threat is for real.
While nearly all darklords have at least some control over their domains' environment, the batshit-insane Easan the Mad cranks this influence Up to Eleven, such that the landscape of Vechor shifts with his mood swings and he can physically alter anyone who lives there just by thinking about them. Another insane villain, Davion, causes drastic changes to the village where he lives, every time he swaps between his four personalities.
In the 3rd Edition of Dungeons & Dragons, a cleric with access to the Madness Domain (offered by Tharizdun and a couple of other evil deities) gains a Insanity Score equal to half his class level. This lets them add this score to his Wisdom Score whenever Wisdom applies to spellcasting. However, it has a disadvantage: for anything else that involves Wisdom he subtracts the Insanity Score from his Wisdom Score.
The Madness Domain also gives the cleric the ability to "act with the clarity of true madness", meaning he gains the above benefit to any roll involving Wisdom, but only once per day (and he must decide to use this effect before making the roll; if it fails anyway, it still counts as used).
There is also the Tainted Scholar prestige class, which gains more spells and greater spell potency, as well as some other benefits, based on how insane they are. It is considered one of the more powerful prestige classes in the game.
People known as "The Lost" from the obscure RPG "In Dark Alleys" have such a bad direction sense that it defies time and space. By wandering randomly through streets and corridors they can, in about 15 minutes, get anywhere. The other side of town? They could go there. Another continent? yes. Your baby brother's subconscious? Yes, and they'd probably enjoy it, since most of them are either complete Cloud Cuckoo Landers or constantly high/drunk. The same stupidity/clumsiness/insanity also allows them to pull anything out of a bag or pocket. Some people can just casually accept the fact that there could be, say, a hand grenade in their wallet they don't remember ever having put in there.
Dont Rest Your Head, a system where you become more powerful the longer you haven't slept. Indeed, your insomnia grants you "madness powers" which can be used to warp the world around you as you see fit. One of the examples in the sourcebook is the madness power of preparedness, where, if you activate it to a certain degree, you can have "conveniently" happened to have picked up the combination to a vault you are just now getting to. And then there's probability.
This is the case for decks built around the keyword abilities Hellbent, Madness (yeah) and, to a lesser extent, Dredge. Sanity is represented by the cards left in your hand and in your library; an empty hand is unstable, an empty library is when a planeswalker is going to completely lose their mind. Madness allows you to sacrifice short-term sanity to play the card you're discarding cheaply; Hellbent denotes cards that gain an advantage when your hand is empty; and Dredge allows you to affect your long-term sanity to recur things from your graveyard.
The general idea behind the Cabal's dementia magic. Some willingly give in to the madness to attain greater power; only the true masters can maintain their sanity.
In KULT, this is a way to see through the illusion that is our reality. If you proceed long enough, you may become the immortal superbeing every human really is, but you risk to be swallowed by the Achlys.
City of Heroes plays this straight at moments, and subverts it as well. Rularuu's Lanaru aspect broke free of and became powerful enough to frighten Rularuu himself thanks to Lanaru's remarkable insanity. On the other hand, the Clockwork King is undeniably and completely insane, a Mad Scientist, but one of his particular delusions — that he can create working robots out of random junk — prevents him from realizing his true abilities in our universe.
He's actually a godlike psychic, animating every Clockwork in the city like a puppet through the power of his fractured subconscious. Detailed examination of Clockwork reveals that they're just collections of parts that couldn't possibly work, and they don't... they function because the Clockwork King believes they should. One of the alternate universes you visit in the late game is a world where the King DID understand his true power. He and his Clockwork are the only ones left alive.
The Suffering has some fun with this. The main character has suffered blackouts his whole life, and in-game develops the ability to trigger those blackouts and turn into a hideous monster. There's a reason this ability is called "Insanity," though—he doesn't actually change, he just goes into an Unstoppable Rage and tears foes apart, hallucinating that it's not really him.
From the Oblivion, Shivering Isles expansion, and less directly in previous releases, there is Sheogorath, Daedric Prince of Madness. In this case the power isn't born of his madness, but is in fact the primordial essence of madness; Sheogorath himself is essentially just an avatar of a larger, immortal force which periodically replaces the avatar with another.
Sheogorath: Now get out of here before I change my mind... or my mind changes me.
One of the possible character backgrounds in "Arcanum", a Victorian Steampunk-meets-high-fantasy RPG, is that you are really an escaped asylum lunatic. Because you are deranged, most people have a lowered initial reaction to your appearance, but you are also so crazy that you take 10% less damage from everything.
Parsee Mizuhashi, the bridge-hauntingGreen-Eyed Monster of Touhou: Subterranean Animism, is explicitly stated in-game to have powers fueled by her jealous rage. There's a reason it's called the "Bridge People No Longer Cross".
In Alpha Protocol, Steven Heck (don't call him "Steve") is one of the world's most talented secret agents. He's also insane. Although never stated outright, there are some suggestions that Steven doesn't even work for the CIA like he claims, but is just some crazy guy who decided to become the world's greatest operative on a whim, and now is.
Portal has Doug Rattman, a programmer at Aperture Science with paranoid schizophrenia who was the only employee that saw GLaDOS as a serious threat to everyone. His schizophrenia manifested in the Companion Cube talking to him and giving him sound advice as he traversed the Aperture Science testing chambers. When he actually takes his meds, he becomes less capable, even getting clipped by the turrets.
In the No More Heroes series, Travis Touchdown is quite out of his mind with the occasional forth wall breaking. It's in the second game that it becomes a power, using his sexual complexes and anger to fuel him in battle and enter super forms, including turning into a tiger. Being sexually attracted to moe designed characters and having a recently killed friend does some weird things to a man.
In American McGee's Alice it is strongly implied that the reason Alice is able to survive and fight is because she's crazy, despite the fact that her goal is to become sane. Basically, it kept her from being completely consumed by madness, which the horrors of Wonderland would have quickly done to a lucid person.
Might and Magic: Heroes VI has the Maniacs and the Demented, demons created to act as servants to the Demon Overlord of madness. Their thoughts are so alien to other beings that they are completely immune to mind-affecting spells, and fighting one can be a disconcerting experience, since they howl as though they're in pain whenever they attack someone, and laugh maniacally in response to being attacked.
8-Bit Theater has very many characters that work like this. Let us categorize them:
Mad Genius: Red Mage, a Munchkin who believes that the universe runs on Tabletop RPG rules. Considering the fact that he has survived certain death TWICE by not writing down the damage he took it is very well possible that this is true. As far as RM goes, at least... And skeletons are wholly vestigial, anyway. Everyone knows that... However, whenever RM is trying to come up with a plan, as usual completely detached from logic, realism and the fundamental physical laws of the universe, this backfires, because reality doesn't work that way! ... Except when it's funnierif it does.
At least once, Red Mage tries to invoke this deliberately. Logical plans are flawed, he tells Black Mage, because of their very reliance on logic. A failure at any given step causes the whole plan to fall apart. By formulating a plan that is completely detached from logic, he has actually insulated it against such failure; since no one step actually relates to any of the others, screwing up won't have any effect on the overall plan.
He also doubles over as something of a Mad God, since he is older than the entire universe itself, borderline omniscient, and is, as mentioned above, an Ax-Crazy, SociopathicJerkass... Earth used to have 36 hour days, but Sarda keeps them at 24 hours just to make everyone hurry.
Reality Warper: Though they probably are more stupid than crazy, both Black Belt and Fighter shows signs of this category. E.g., Black Belt has, due to his No Sense of Direction, on numberous occasions altered the space-time continuum in ways that shouldn't be possible. And when their deathtrap of a submarine was filling with water, Fighter was able to return it to normal by pulling out the plug and letting the water drain out.
Thief: I hate it when he makes things that don't make sense make sense.
Achewood features this in the Great Outdoor Fight. Those with Blood of Champions are known for entering fugues which destroy their minds afterward. Ray succumbs to this at first, but gradually channels it into a productive "berserker" state, which according to Roast Beef is the difference between "a lunatic [and] a pissed man with goals[.]"
Girl Genius does this with Mad Science: Sparks have manic phases where they're deeply intuitive, focused, driven, and scientifically brilliant, although they might forget to eat for days... or, you know, to not be evil.
Carson von Mekkhan: In my experience, a strong Heterodyne will take about two hours to truly warp laws of nature
There's also a subtle inversion in the fact that while all Sparks go crazy at times, it's actually the ones who can stay in control while "in the madness place" who are the most dangerous people in the setting, because they can direct and channel their insanity. Baron Wulfenbach, his son Gil, and Agatha, are the best examples.
In Mortifer, the less connections a demon has between his mind and reality, the more of their power they can access (this only applies when they're on earth, though).
It isn't the main source of his power, but Xykon in The Order of the Stick definitely seems to be more powerful thanks to being completely mad. Inscribing a symbol of insanity onto a bouncy ball and throwing it into a room full of people isn't the sort of tactic a sane sorcerer would come up with.
Monster Clown and maddest of the mad scientists Rhyme in Star Harbor Nights is able to produce feats even other mad scientists can't duplicate (like making helium burn like hydrogen) because she follows "an alternate ordering of the universe"
Doctor Steel. "Being insane allows me to be who I want to be. There are no longer any rules. I am crazy, and I will become World Emperor. Even if it's simply because I'm convinced I'm World Emperor." "You can get away with pretty much anything if you're bonkers."
Most of the Channel Awesome reviewers believe this applies to Dr. Insano in Kickassia. First the Nostalgia Critic attempts to get Spoony to let him out, followed by those rebelling against him. Only Linkara sees that this is a dumb idea, but the others are convinced that you need a madman to stop another madman. Sure enough, Insano gets his ass kicked, although he comes closer to defeating the critic than any other individual rebel.
In Worm, power is born more from psychological trauma than from madness. In order for a potential parahuman to unlock his/her latent powers, he/she must experience a "trigger" event that causes extreme stress and pushes the person to the breaking point. This may explain why there are more villains than heroes in the Wormverse, and the heroes tend to carry around a lot of psychological baggage.
The Transformers Animated villain Blitzwing seems to be able to use two elemental weapons and two vehicle modes due to his Split Personality. However, it's not completely clear if one causes the other.
Word of God says that the experimentation that turned him into a triple changer made him nutty, not the power itself. Knowing the usual Decepticon "research process", it's hardly surprising.
Though it's worth noting that while his "German Pilot" personality can only use the plane form and ice weapons and the "Schwarzenegger" personality can only use fire and the tank form, the "Nutty" persona can use all his abilities at will.
Nightmare Moon seemed to be this - at least Princess Luna seemed much weaker once she'd regained her sanity. However, when she returned in the second season, we find out that she is in fact so powerful as herself that she can be destructive without meaning it if she doesn't work to keep a lid on her godlike powers. She seems more powerful than her sister because she is less restrained, and of course being evil is the logical extension of this.
Rainbow Dash: (After watching Pinkie Pie slow her descent mid-air to avoid a splash) How did you do that?
Pinkie Pie: Just trying to show a little consideration for my sleepy friend.
Viking berskers demonstrated this trope. They would fly into violent psychotic rages that would essentially remove the brain's natural limiters on their muscles, increasing their raw strength several times over, granting them what resembled superhuman strength (remember that these were men who ordinarily could row a longship for hours or days, and then still put on chain armor and pick up a shield and war-axe and fight, and were just flat-out bigger than a lot of European peoples) and grant them an immunity to pain (the adrenaline and anger would block it out), making them resemble super-human engines of destruction. There is some debate over whether or not these rages were induced by hallucinogens or just a natural tendency of the people to get really really pissed. Or both.
These rages have also been said to make them immune to "blade and fire", which holds up at least somewhat: while both of those things hurt a lot and can kill you, in most cases (barring decapitations and other instant kills) it's really the pain that puts you down in the short term—and as stated above, pain is sort of a secondary concern for someone in a berserking rage—so blunt weapons are more effective, as the injuries caused by them tend to actually break the machinery of the body rather than just hurt a lot.
The legendarily bad-tempered Egil Skalla-Grimsson was an example of a Norse warrior who really did fly into a rage over nothing. He has recently been determined to have had Paget's disease. This disorder causes the bones to become very thick and strong, but also causes circulation problems, deafness, blindness, and various minor deformities. Quite significantly, it also causes chronic pain that prevents the victim from sleeping properly, ensuring a chronic bad temper.