Light uses this technique in Death Note by having Misa and himself give up control of the Death Note to another individual to throw L off his track and clear his and Misa's names. The plan is uncannily successful, and ultimately L is killed as a result.
"Just as planned!"
In the movie, however, things go differently: L figures out what happened the second he gets his hands on Light's note and sees the rule about losing memories. Seeing Light's plan, L plays along, writing his own name in a Death Note (making him immune to death by any other Note for 23 days), and catches Light in the act.
What's particularly notable about the manga example is that technically the person whose actions Light is anticipating is himself. And it's not even really himself, but rather what he would have been had he never found the Death Note. In other words, he has to anticipate just what he would have done and how he would have acted had he never found the Death Note, do the same process with Misa, work in how L and the gang would react to that, and then work in how the as of yet to him unknown corporate executive is going to act, and then figure out how everyone's going to react to that.
He also knew that if the plan had failed, he would have been safe and lived, although not as Kira, because Higuchi would be caught and blamed for everything. Light would then be off the hook and live his life out normally... though, as the movie shows, even that wasn't without a failure condition.
L staves off many episodes using this technique, in fact his entire operation in the first half could probably be summed up as this. Serial killer eliminating prisoners all over the world through methods unseen, unheard, and untraceable? Set yourself up on television with a substitute so that Kira will attempt to make a move against you. Before he does that, make sure the broadcast is isolated in the single region where similar killings were determined to have started. Thus, since your substitute was killed in that area and was killed in the same way as many others, that person must be in that area, watching TV at some point, and must only be able to kill with a name and a face. Needless to say, Light is thoroughly whipped by this.
The next part of L's operation also did this. There was no way to witness Kira physically, but L made extreme observation of his killing patterns and even formed a "personality profile" off of it. When he whittled this down to Light Yagami, he decided to meet Light in college. And how did he introduce himself? Telling Light that he was the detective hunting Kira, and knowing subconsciously that there was no way Light could do anything about it. He then matches Kira's characteristics - childishness, poor sportsmanship - with Light's regular behaviors. As the investigation winds down and lines have been drawn, L even delves into Genre Savvy by warning his investigation team that should he be killed within a certain timespan, they should follow his original assumption that Light IS Kira.
IzayaOrihara of Durarara!! oozes this trope like a fountain. He has stated that it is solely for his own amusement, but there is no denying that his manipulative capabilities over the residents of Ikebukuro are amazing. Few have been able to work their way around his plans.
The only person Izaya opts to work his way around is Shizuo, whose combination of Super Strength, intense hatred for Izaya, and being Too Dumb to Fool make him a very dangerous potential Spanner in the Works who Izaya will go to great lengths to keep otherwise occupied with relentless trolling.
The character Joseph Joestar in part 2 of the manga JoJo's Bizarre Adventure states that him drawing the character Cars to be on top of the volcano when it erupts in the final battle was all part of an elaborate battle strategy. However, this is quickly thrown out the window seconds later when Joseph's inner thoughts reveal that Cars standing there when the volcano erupted was just a coincidence, and the Joseph simply didn't want it to look like he only won due to a fluke.
Jotaro is good at this. He showed us how by outwitting Daniel J D'arby in a high stakes with souls on the line. Jotaro not only caught D'arby off guard with various call outs, but raised the stakes higher than D'arby could believe, eventually coming to a head when D'arby subconsciously admitted defeat.
The Major pulled one against Alucard in Hellsing. Recognizing Alucard's and his master's style, he fabricated an emergency by having several traitor vampires slaughter the entire population of a VTOL warship, and sent one of Millennium's core lieutenants there to apparently trap and kill Alucard when he was sent to deal with the situation. Instead, everybody in the ship was to be the trap to lock him in the destroyed warship when he inevitably slaughtered the entire squad, as vampires cannot cross water. While Alucard used his powers to psychically drag the damaged ship to the Thames, the Major's army had already slaughtered London and forced a Catholic-Protestant-vampire war in the burning city, making him too late to stop it.
It could even be argued that the whole plan of the major revolved doing it at the very end of the series by expecting Alucard first to absorb the life of Officer Schrödinger and then to release Level 0, thus 'poisoning' him, which it happened...everything else only served to lead to that moment, that assumption, fifty crazy long years planning on this greatest Batman Gambit.
The entire plot of the Ghost in the Shell movie is one big Batman Gambit by an opponent aptly called the Puppetmaster. At one point he hacks into the heavily secured factory that builds the replacement parts for the Major's and Batou's cybernetic bodies and had it create a unique cybernetic body. Then he had the body walk out of the factory and to the next highway to step in front of the next passing truck. As access to the factory would also allow him to temper with the replacement parts for Section 9, they really want the remains of the rogue cyborg body in their lab for analyzation. Exactly where the AI wanted to be, but couldn't get to without his handlers stopping him. Everything that happens in the movie is all part of his plan to get an opportunity to talk with the Major.
Mazinger Z: Kouji Kabuto from Mazinger Z used several throughout the wholetrilogy, manipulating his enemies to make right what he wanted, often mixing them with Crazy Enough to Work. For example, in one chapter a Mechanical Beast -Kirma K5- sliced a chunk of one Mazinger-Z's wing during an aerial battle. Kouji could not balance his Humongous Mecha and he fell towards the ground. Then he goaded Kirma into attacking him again, and positioned Mazinger so his foe's Sinister Scythe sliced a chunk of the another wing. Now the Mazinger wings were the same length again, and his enemy was nearby, he could balance Mazinger back and grab Kirma. As he was beating the crap out of the Robeast he gloated it should have let him drop.
Neon Genesis Evangelion: Gendo Ikari had an elaborate Batman Gambit in motion, manipulating people throughout the entire series to react in ways he expected and behave right like he wanted or needed. However, the success of his plan completely hinged on everybody acting like he had foreseen, so when his most important pawn refused do what he wanted, she completely ruined his plan.
The heroes pull this off when Captain Misamaru of Martian Successor Nadesico successfully predicts the actions of the Jovians and herds their superweapon into a trap, by continually convincing them the trap is elsewhere.
In YuYu Hakusho, the reason Sensui set it up so that the tunnel to Makai would be opened, and the events that came afterwards, is so that he could die in Makai at the hands of a powerful demon, since it would be nobler than to die of the terminal illness he had. Said strong demon happened to be Yusuke, possessed by the spirit of his ancestor, Raizen.
Later in the manga-only final arc, it was revealed that everything the heroes faced was an elaborate setup by King Enma, who purposely pardoned dangerous criminals with the specific hope that his detectives would capture them and thus make himself look better.
In The Prince of Tennis, Badass Bookworm Sadaharu Inui is having his ass kicked in the courts by another Badass Bookworm, his former partner Renji Yanagi, who is as good as using info as Inui himself is. Then, Inui claims he won't use his data anymore... and it turns out this was fake, since he lulled Renji into a false sense of security and shaped their game into a copy of their last unfinished match. With this in mind, Inui ultimately wins.
In general, several of the tennis games in the series use these Gambits as part of the strategy. Examples are Fuji Shuusuke (who let his brother's team manager from Saint Rudolph almost beat him, only to come back with a vengeance and CRUSH him as punishment for mistreating Yuuta) and Kawamura Takashi (who used on purpose two variations of the Hadokyuu to weaken Kabaji in the Hyoutei arc and force him into a draw) from Seigaku, as well as Masaharu Niou of Rikkaidai who switched places with his partner Hiroshi Yagyuu for half a game to trick both teammates and rivals into believing one was the other, confusing Eiji and Oishi and defeating them.
Lelouch, the protagonist of Code Geass, does this a lot. Along with Lelouch, Cornelia and Schneizel also have been known to pull less convoluted Batman Gambits of their own. It must run in the family.
In the second season, Lelouch pulls this stunt again, against Schneizel who really should have known better, and with a much more improbable level of precision.
When Mao came back, he used his own power of absolute command to give himself orders to tell Suzaku to charge in guns blazing when he heard Lelouch scream, then wiped his own memory of this. Lelouch then plays right into Mao's hands, seemingly completely defeated. When he screams, Suzaku comes in and the gambit is revealed. It's hopelessly convoluted and yet masterful.
When Lelouch, as Zero the masked terrorist, claims that he wishes to join Nunally's Special Administrative Region of Japan along with his Black Knights, he later makes a secret pact with the Britannians to have him pardoned for his crimes and be exiled safely, which allows the Britannians to attempt to engineer the situation in their favor: they would broadcast that they were pardoning Zero for all crimes at the opening ceremony, and Zero only, therefore meaning that the other members of the Black Knights were still accountable for crimes, and were thus betrayed by Zero. The Britannians had hoped that this would trigger a riot amongst the betrayed Black Knights, which would provide them with ample excuse to put them down with lethal force. Problem is, Lelouch had earlier made the point that Zero is an ideal, not a man. So, when everyone at the assembly suddenly puts on Zero masks, no one can try to force the issue. The only person who knows his identity, Suzaku, can't call him on it. Thus, they are all granted immunity and a safe trip out. It heavily relied on Suzaku being sentimental enough to stop the other government officials from opening fire on the crowd, and it came within a hair's breadth of backfiring.
Lelouch often has plans that rely on the fact that while "other" people have a sense of honor, he does not, and uses it to trip them up. Good examples include the way he tricked Guilford into a duel of honor, and then pulled a sneaky trick on him. There's also him using the UFN leaders as hostages to trick the Black Knights into preventing Schneizel from nuking his ass, forcing him into conventional strategy.
Of course, this is more along the lines of other people believing that he doesn't have a sense of honour, when he really does. Of course, manipulating the battlefield he and Guilford was on was indeed a very sneaky trick, it's nothing that shouldn't have been expected. And it was a hostage situation, after all, so Guildford was the first to use a sneaky trick. And he certainly wouldn't have harmed the UFN leaders, even though he had weapons trained on them - at least, not the two young girls he's familiar with.
The UFN example is doubly appropriate when you realise that Schenizel does know that Lelouch won't hurt the UFN Leaders. That's why he has Nunnally at his side, and has her accuse Lelouch - by doing so, Lelouch becomes incapable of using any FLEIJA weapons that he may have, for fear of killing Nunnally, and Nunnally fails to realise that's what is happening, and is willing to go along as a result. Of course, Lelouch never had a FLEIJA in the first place, but he may have considered the idea as a more appropriate alternative to creating the FLEIJA Canceller. In fact, Schneizel himself planned to kill the UFN Leaders, as well as the Black Knights. No, Schneizel attempts to fight Lelouch conventionally, because he's certain of his own invulnerability and wants to try confronting Lelouch through wits, rather than brute force, and if can't win, he has FLEIJA as an alternative. And when his patience wears thin, and he does. resort to FLEIJA.. that's just what Lelouch wanted.
An example where it isn't set up by a villain: everything in Card Captor Sakura (except for the Yaoi Guys) is an elaborate plan by Clow Reed to ensure that his magical creations would have the best master (Sakura) after his death. The reason Sakura captures the cards and then masters them is because if she didn't, they'll cause a lot of trouble in her town. However, Eriol (Clow's reincarnation, who kept all of his previous life's memories) made sure she wasn't in any real danger (or anyone else in town, for that matter). This is possible because Clow was prophetic to the point of omniscience, able to see the future so perfectly that he WAS able to see every consequence of his actions and what to cause to happen. Except for the Yaoi Guys, Eriol admits that one surprised him (he had intended Yue/Yukito to have a Bodyguard Crush on Sakura, but Yukito fell for Sakura's older brother instead). As an interesting side point, the whole point of this in the manga was to rid himself of that power, which would require Sakura to be a stronger mage than him.
Of course, there was also the goal of making sure the Clow Cards power supply didn't fail, leading to their deaths, too. They couldn't run on Clow's residual energy forever, but the real danger was when they were left in limbo and Sakura had to master them.
Griffith from Berserk does this against the Queen of Midland and a conspiracy of nobles who wanted him dead. The conspirators apparently poison Griffith, but unbeknownst to them, Foss, the key figure in the conspiracy to kill Griffith, is blackmailed by Griffith into putting a drug meant to give the appearance of death in his cup instead of poison. The Queen and her nobles, believing themselves to be victorious, all gather together in celebration at her castle, and that's when Griffith springs his trap, having them all locked inside the castle and then having the place set ablaze, killing them all.
In Kure-nai, the protagonist Benika pulls off such a gambit with her "kidnapping" of Murasaki Kuhouin. Although she assigned Shinkurou as a bodyguard to protect Murasaki from being recaptured by her own (very messed up) family, she never expected to keep her safe from them forever. Instead, her plan was to "poison the prize": by letting Murasaki experience enough of the outside world while living with Shinkurou, she would hopefully return to the Kuhouin family with the knowledge and willpower to bring down the family's traditions from the inside.
Gunnm: Caerula Sanguis applies a Batman Gambit in a rather unorthodox way: one-on-one battle. In Volume 6 of the Last Order storyline, after a token unarmed exchange with Alita and a cloud of throwing knives, she announces a special technique called the Eight-Block Death Gate Array. In the beginning, it looks merely like a fast combination of sword strikes. However, this is only the beginning of the ruse. The Array is actually not a technique or even a combination, but a tactic custom-tailored to every single enemy she meets. By using her 700 years of combat experience and her ability to sense neural pulse flow and thus infer the intentions and thoughts of her opponents, she takes the terrain as well as the opponent's capabilities, reflexes, and even their personalities all into account, shaping the match such that all her opponent's actions, both conscious and unconscious, lead them inexorably to a location where they will be completely helpless against her full attack. She is apparently good enough that she can suss this out completely within the first few seconds of any given fight. The only ways to defeat the Array in a duel are to be completely certain you are doing one thing while doing something completely different (such as freezing when every cell in your body is yelling for you to run), or by completely banishing one's ingrained reflexes, becoming completely aware of the self and the world, and thus able to make true choices.
Jinnai, the Smug Snake of El-Hazard: The Magnificent World, makes a bid for Magnificent Bastardy with this one: There's a Forgotten Superweapon that could help his Bugrom allies win their war, but only the good-aligned Priestesses of Mt. Muldoon know where it's hidden away. So he orders the Bugrom fleet to set sail. The heroes, thinking Jinnai knows where it is, panic and sail off with the priestesses leading the way... and Jinnai simply follows them to the right island.
Aeolia Schenberg's feats in Gundam 00 are oft barely noticed. In spite of living 200 years before the events of the anime, he managed to lay down a framework for Celestial Being and created an A.I. that would be able to create missions and postulate possible strategies for the group to carry out. Not only that, he theorized and developed technologies that were to come to be centuries beyond his time, including the Twin Drive for the Gundam 00 (and the solar furnaces themselves for that matter) and the orbital elevators. Although he planned to go into cryogenic sleep and wake up in the future, he also planned for his death. He made sure that if he was ever killed, the Trans-Am system would be activated in all Gundams, as well as making other information stored outside of Veda accessible (like the aforementioned Twin Drive). Aeolia's surprise 200-year old plans often manage to foil Ribbons's, the resident Magnificent Bastard, own machinations, to which he frequently expresses frustration at. After all, he thought he was just carrying out Aeolia's wishes too.
Federation soldier Hank Hercules initiates a coup de'tat and seizes control of the Africa Tower elevator's low orbital station, counting on a lethal response from the A-LAWS that will illuminate the citizens into their brutal ways. However, the A-LAWS use the incident to facilitate their own Gambit, using their information control to frame the mutineers for the A-LAWS' ruthless attack on the citizens and the destruction of the tower itself, all to completely discredit the mutineers and their anti-government allies, while giving the A-LAWS a pretext to consolidate the full might of the Federation army under their banner.
Though this turns out to have fatal consequences later on, as some of the more important members of A-LAWS make a Heel-Face Turn in response.
Xellos does this all the time in Slayers, which is not particularly difficult, as everyone in that series is extremely predictable. Usually, whatever he does serves his plans in some way, but a lot of the time he just manipulates them because he thinks it's funny. It generally is.
Shikamaru Nara from Naruto won most of his early major battles by using Batman Gambits to manipulate his opponents into just the right spot such that he can either catch their shadows or force them to harm themselves through use of this otherwise harmless technique.
Deidara also pulls one to capture Gaara: he realizes the sand Gaara is attacking with is the one he has the most control over, so when Gaara rips Deidara's arm off he has the living clay inside said arm crawl inside this sand. He then sends a huge bomb towards the Sand Village so Gaara has to send most of his sand to withhold the blast, forcing Gaara to block his next attack with his best sand, has the clay he left inside that sand crawling into the center of Gaara's sand sheld, then detonates them.
Fullmetal Alchemist — Father's entire plan coming to fruition rests on the fact that he and the homunculi knew that the sacrifices would never back down and leave the country.
Dr. Marcoh also pulls one when he springs a trap on Envy by having Reverse Mole Zanpano call him and give him Marcoh's location. Marcoh knew Envy was so cruel and ruthless that he would personally come down to screw with him even more.
And the actual goal of the entire plan is entirely based on this as well. The targeted populations have advanced knowedge of alchemy, including methods for creating a Philosopher's Stone. The villain is relying on somebody to use that knowledge and manufacture a Stone, which can then be stolen.
Also in the 2003 anime, more than once Colonel Mustang orders Edward into a situation where his desire for justice causes him to go off mission and break the rules. Mustang chastises him for it, but Edward's good deeds always turn out to be very convenient for Mustang...
In Dragon Ball Z, Vegito pulls this off on Majin Buu. It seems that Vegito can beat Buu at any time, but is just toying with him, while the Kais are asking why he just doesn't Get It Over With. Turns out Vegito is purposely making Buu angry enough to get desperate, which is when Buu starts absorbing people. Vegito allows himself to get absorbed on purpose so he can rescue Gohan, Trunks, Goten, and Piccolo, who were absorbed earlier.
He clearly gets this from Vegeta, who also pulls a couple Batman Gambits throughout the series. In the Namek saga he steals all of Frieza's dragon balls while both Frieza and Zarbon are in the same ship as he is, by fooling them into thinking he'd already left the ship.
Piccolo pulls one off against Imperfect Cell by tricking him into revealing who he is, how he was created and what his plan is by making him think that he is going to die after having all the energy in his right arm drained from Imperfect Cell.
Akiyama Shinichi of Liar Game uses this trope pretty much continuously. I mean, just look at the way he plays on Fujisawa's anxiety in the first round through constant surveillance, panicking Fujisawa enough for him to believe that Akiyama was the LGT representative arriving to collect the 100 million yen. Yes, Fujiyama handed his money to his opponent after guarding it so intensely for 30 days.
A bigger example of this can be found during the musical chairs arc, in which Akiyama predicts and manipulates the extras, a traitorous teammate, as well as the enemy teams into making a 'trash medal' player win in the end. Had anyone gotten suspicious at any point, had the enemy team decided to sacrifice the trash medal player early, or had the team in particular he'd gathered the medals for lost out to the other, it would have all fallen apart, making it a wonderful example of this trope.
Both Marik and Dark Bakura in Yu-Gi-Oh! tend to operate this way. One notable example is Dark Bakura's duel with Yami during the Battle City finals. When Yami secures certain victory by playing his god card, Marik's gambit to to release Bakura from Dark Bakura's control, and thus force Yami to lose the duel in order to protect his critically ill friend's life, has two flaws in it. The first is that Marik didn't expect Dark Bakura to protect Bakura by re-possessing him. The second — and one which everyone seems to overlook — is that Yami could simply ask Bakura to forfeit the duel.
For that matter, most of the card game duels played in this series operate on Batman Gambits. Duelists continually bluff and trick their opponents into helping them set up an elaborate ploy that will secure them certain victory, unless their opponent then spots the tiny but fatal flaw, or just draws the right card out of dumb luck.
One of the reasons that Yami Yugi is never really beaten is because he always seems to remain two steps ahead of his opponents... even when they are two steps ahead of him. For the matter, one can probably rename it "Yami Yugi Syndrome".
In fact, Raphael, the only duelist on the show who can claim to have defeated him fairly (except Yugi himself, in the finale), did so by using a Batman Gambit of his own.
One of the most excellent examples occured in Yu-Gi-Oh! 5Ds, during Yusei and Jack's second duel in the final duel of the Fortune Cup. Jack attempted a Batman Gambit using his Speed Spell - The End of Storm card, expecting Yusei to use the same trick to counter it that he did in their first duel. Unfortunately for Jack, Yusei was using a Batman Gambit of his own, and had wanted Jack to try that; his true way of countering it this time took Jack by complete surprise.
Another good example was Tron in Yu-Gi-Oh! ZEXAL. He came off as a sadistic monster, cruelly taunting his opponents and abusing his own henchmen. While it would be wrong to say he didn't mean any of it (he truly was a sadistic monster), he did have a reason for doing so that he eventually revealed: His most powerful card was Number 69: Heraldry Crest. This incredibly powerful Number was described as the "embodiment of rage" and very difficult to control; using it required vast amounts of anger. So he simply did his best to make everyone he encountered - minions, enemies, and Unwitting Pawns - as angry as possible, enabling him to control his ultimate weapon with ease when he finally unleashed it.
Eda in Black Lagoon has a very crude (but effective and hilarious) one set up that she uses to bushwhack "Greenback" Jane. Even the other characters think the plan is stupid.
Hansel and Gretel managed to divert the majority of the hiredguns out for their bounty with one of these. Step #1: get a car with tinted windows so no one can see who's inside. Step #2: Bribe two local orphans to pose as you by putting on your clothes and wigs similar to your hair, then get them in the car and have them drive off. Step #3: Laugh as the stupid bounty hunters blow the car to smithereens and realize how they've been had by two little kids.
Rock attempts one that would bring Ronapour to the world's attention in the "Roberta's Blood Trail" Arc by manipulating or predicting the actions of just about everyone in the city. And he would would have gotten away with it too, if he'd known about the undercover CIA agent that he practically told his entire plan to.
Though he frequently makes use of the Indy Ploy to succeed, the titular character of Ranma ½ is not adverse to a good Batman Gambit, particularly in the manga. Though they do usually fail and force him to revert to Indy Ploy, that's because Ranma Saotome's "friends" and enemies are extremely unpredictable, and so prone to reacting in ways other then what he expected. The best example would be the first Pantyhose Taro story, where he sets up an elaborate plan intended to convince Dirty Old Man and Jerk AssHapposai to rename Taro by giving him a faked dream sequence in which Happosai sees a future in which Taro has, due to his name, become a terrible Panty Thief, and is then sent back to the day he named Taro. Ranma expects Happosai to choose to give Taro a new name. Instead, Happosai proves his evil credentials by promptly trying to murder what he thinks is baby Taro with his bomb attacks.
In Trigun, Vash wins most of his battles by quickly and carefully studying his opponent's personality and mannerisms to predict how the opponent will act and move.
In Katekyo Hitman Reborn!, Tsuna, or at least his future self from ten year's time, pulls this off to the exteme in order to save multiple universes from the arc's BigBad Byakuran. But mainly for training himself and his family up.
In the anime-only Zanpakutou Unknown Tales arc, Muramasa pulls on one Ichigo. Knowing that, as the hero, Ichigo feels the need to rush in to protect his friends and allies, using full power, Muramasa sets up a scenario whereby Ichigo has to do exactly that to save Yamamoto who has been trapped by Muramasa behind a barrier since the start of the rebellion. The truth is that Yamamoto had set up the barrier to keep Muramasa away from and wasn't trapped at all. Muramasa needed Ichigo's power, combined with his own, to break Yamamoto's barrier and gain access to Yamamoto's zanpakutou. Murmamasa had even gone to the extreme of ensuring that any shinigami travelling with Ichigo would be isolated from him along the way so that they couldn't work out Muramasa's plan and warn Ichigo. As it was, those shinigami only worked out the truth at the last minute, and only because Muramasa going to the trouble of isolating them from Ichigo clued them in to Muramasa's real plan. They were therefore unable to warn Ichigo in time and Muramasa's plan succeeded.
In the manga Aizen kidnaps Orihime because he knows that Soul Society knows he'd be interested in Orihime's unusual power. Although Soul Society's plan was to leave Orihime alone until they could plan a proper invasion of Hueco Mundo, Ichigo impulsively rushes off with his friends to rescue her. This is exactly what Aizen hoped for. The result is that Soul Society is forced to send in four captains earlier than planned, at which point Aizen shuts the door behind them, leaving Ichigo's group and four captains stranded in Hueco Mundo. This allows Aizen to confront a vastly reduced Gotei 13 force at Karakura Town.
In School Mermaid, the one pulling the strings is Haruko, who deliberately brought the unconfident, nonathletic Yoshiko to hunt mermaids while ensuring she wouldn't find out the consequences of failure until it was too late: being transformed into a mermaid herself. Turns out she needed a mermaid whose name started with the same letter in order to make her crush fall under its spell... Guess who qualifies.
In Pokémon Special, Fantina has a pretty good one set up as her Gym gimmick. First, the challenger has to solve some difficult but straight forward math questions to reach her, forcing the challenger to adapt a logical way of thinking. When the challenger actually reaches her, her Pokemon's illusions kick in, totally destroying all that built up logic and with it the trainer's brain. If a less logical person were to reach her (presumably just by defeating all the Gym Trainers instead of answering the questions), he would just be caught up in the illusion and not realize he's getting the crap beaten out of him.
In One Piece, Blackbeard successfully defeats and captures Ace and hands him over to the Marines, earning him the title of Warlord of the Sea. Knowing Whitebeard's familial protectiveness,a chain of events would led up to a war between the Whitebeard Pirates and the Marines at Marineford, which gave him a clear opening to break into Impel Down and recruit the most dangerous and powerful prisoners into his crew.
But the gambit doesn't stop there; after that part's success, he takes his crew to Marineford, where he knew Whitebeard would have been worn down from taking on the World Government's full force. He then has his entire crew attack Whitebeard, finally killing him, and then proceeds to steal the power of the Tremor-Tremor Fruit from him. And as if his threat level wasn't already Up to Eleven with that move, he then proceeds to usurp Whitebeard's territories and titles, both that of the Strongest Man in the World and that of one of the Four Emperors. The bottom line? A monster pulled off a successful Batman Gambit, and has officially become the series' Big Bad.
In Revolutionary Girl Utena, most of Touga (and by extension, Akio) and Mikage's plans to obtain the Rose Bride rely on these. In particular, there's Episode 11 (counting on the fact that Utena would prioritize her feelings for Dios over her desire to emulate him, costing her the duel), all of the Black Rose Duelist arc (Mikage sets up emotional catastrophes in the lives of students so they'll enter his seminar), and Episode 38 (which relied on Utena prioritizing Anthy over Akio... not only setting her up for a stab in the back, but cementing her as a prince who can be used to open the Rose Gate.)
Episode 9 of Puella Magi Madoka Magica has a spectacular one pulled by Kyubey on Kyoko and Homura. Anticipating that Kyoko's re-emerging heroic tendencies will lead her to try and change Sayaka back from witch form, he vaguely implies that Madoka could use The Power of Friendship to do so (and of course, Kyoko would need to accompany her for protection). Of course, it was an impossible prospect from the start, and not only does Kyoko pull a Heroic Sacrifice to defeat the witch and save Madoka, but it leaves Homura with no other Puella Magi to help her battle the upcoming Walpurgisnacht - which practically forces the incredibly self-sacrificing Madoka to make a contract, guaranteeing her transformation into a witch that will destroy the world and provide Kyuubey with endless amounts of energy.
It backfires spectacularly in the final episode, when Madoka pulls a gambit of her own against him. While Madoka does indeed make a contract with Kyubey, the terms of said contract are that witches never exist in the first place. The end result? Madoka becomes a Magical Girl goddess and changes the universe so that Kyubey has to rely on a much less efficient form of energy, along with altering the little bastard's memory so that he doesn't even know what a "witch" is. Oh, and she binds it to the new system so he won't be evil anymore. Ironically, his ultimate plan fails because of anotherHeroic Sacrifice—one he did not expect.
Word of God says Kyubey was never evil in the first place. Just completely logical. On top of that, Kyubey didn't say anything about Madoka being able to save Sayaka through friendship. Kyoko asked Kyubey if there were a way to save Sayaka from being a witch and it states that if there were, then it hadn't heard of it. Madoka then proceeds to create a balanced system wherein everyone has the maximum amount of happiness that they could have while being in that system.
Keima Katsuragi of The World God Only Knows enacts these regularly, thanks in no small part to his (Dangerous) Genre Savvy, using his knowledge to capture runaway spirits, goddesses and subversive demons. Later, he had Fiore out herself as Vintage's agent and recaptured her several chapters later after she gets away from Haqua and Nora.
Kotetsu pulls off a good one in episode 13 of Tiger & Bunny to get around Jake's mind reading powers. Since he's a bit of an Idiot Hero, it even fools the audience at first.
Hajime in The Kindaichi Case Files does this a lot in order to corner a suspect or find out details that would normally be overlooked including but not limited to: rigging watches, lying about lost items and faking deaths.
In Eyeshield 21, Hiruma, for most of the time, always had some plans that enables him to pull a comeback move from dire straits for his team. Some of his plans often seem ridiculous or just plain batshit insane, but it works most of the time.
Windaria: Lagado turns Alan into his Unwitting Pawn by exploiting his desire to protect The Valley. If Alan said 'We're moving' instead, the plan would have collapsed then and there.
Inukami!: One episode featured the inukami plotting to take advantage of Keita's perverted tendencies, sure their plan would work immediately. Except he showed restraint, foiling their plan.
Yawara: Jigorou has proven to be quite good at these especially when it comes to manipulating his granddaughter Yawara.
Katsuhito Masaki is the main reason that Tenchi netted so many beautiful alien women. Just that it worked a little too well - he got Tenchi to be curious enough about the mysterious cave that he would be able to go in and investigate, freeing Ryoko, which would catch Ayeka's attention (which he figured would happen) and bring her to Earth so that he could send Tenchi back in his place. Nothing in his plans, however, would prepare him for the other four girls who would show up.
This is pretty much the basis for Gamble Fish in which the hero Shirasagi pulls off multiple gambits, sometimes right when he's in the middle of one of his cons or challenges, to ensure his opponents lose their advantage.
In one example, Shirasagi plays a high stake spoker game against a magician who is a master of dealing and swapping out cards to ensure she always gets a winning hand. Shirasagi constantly loses all his money until eventually he has one last thing to bet: His own finger. He loses this round, and the finger gets chopped off with a chainsaw. But it turns out this was his master plan all along, as he is able to use his bloody hand to mark the cards, ensuring his opponent is unable to use her card tricks and forcing her to forfeit.
In Shiki, resident badass and vampire expert Toshio Ozaki pulls one of these on Chizuru, "matriarch" of the vampiric Kirishiki family. Among their many vampire powers is the power of suggestion: the first vampire to bite a victim can instruct that victim to do anything. Up until this point, the vamps have used this power to prevent their victims from seeking medical attention from Ozaki, who is the only doctor in the village. But because Ozaki, who experimented on his vampire wife, now knows all their weaknesses and is a real threat, Chizuru bites him and implants the suggestion that he burn all his notes regarding the vampires and their connection to the deaths in Sotoba. Ozaki carries out her wishes the next day. Later, Chizuru notices the Kagura Festival and expresses interest in it, but can't go due to the religious symbolism. Ozaki convinces her to go anyway, assuring her that if she changes her mind, they can go back. Chizuru agrees, but starts to chicken out when they get too deep into the festival. She orders Ozaki to escort her back to the clinic, but he refuses and reveals a second set of bite marks in a place that Chizuru never bit him. Chizuru was actually the *second* vampire to bite Ozaki, the first being Natsuno, who is working with Ozaki to take down the other vampires. Ozaki went along with Chizuru's wishes to gain her trust, lure her to the festival, and then reveal her as a vampire to the rest of the villagers. It totally works.
In Kurokono Basuke near the end of the final quarter of the Seirin vs Shutoku match, Midorima was about to make a 3-point shot, but believed that Kagami would be able to jump and block his shot once more despite being completely exhausted. So in response he feinted and didn't shoot when Kagami thought he would. Unfortunately for him Kuroko also believed that Kagami would jump and that Midorima would feint so he was able to steal the ball just before the final quarter ended.
In Gundam SEED, Rau Le Crueset's plan to kill everyone hinges on a few Batman Gambits towards the Earth Alliance, namely revealing Operation Spitfire and sending out Flay with the N-Jammer Canceller plans. As expected, Blue Cosmos came out and armed themselves with nukes with Patrick Zala hauling out GENESIS to fight back.
Flit Asuno does a little, but decisive one in Gundam AGE. In the final battle, he realizes that the Vagans are desperately recharging their Wave Motion Gun in an attempt to take down both the Diva and the Gundams. Beacuse of it, Zeheart only manages to destroy is an empty Diva and no Gundams at all at the cost of his loyal dragons, much to his ultimate despair.
As part of being Crazy-Prepared, top assassin Golgo 13 uses Batman Gambits among others all the time. One example is in Episode 42 of the animated television series. Where four assassins are hired to take him out - one of them being female. After being attacked, Golgo 13 kills the three men, but leaves the woman alive. He knocks her out and takes her to a wood cabin. He strips her naked and burns the clothes. He pays a young woman to allow him to use her cabin in exchange for a bright red coat. When Golgo leaves the cabin, the naked woman knocks the young lady out and takes her red coat. She meets up with her boss and they try again to kill Golgo 13. But this is what he wanted: To make the woman lure the mastermind out of hiding, and to make her boss and herself an easy target by the bright red coat she was now wearing. Golgo 13 kills them both with a single rifle shot.
The overarching plot of looking for the feathers turns out to be Fei Wang Reed scattering them and counting on the fact that this will make the group travel universes to find them, thereby supplying him with what he wants: Sakura's body imprinted with the physical memories of different worlds. He even goes so far as to set up the lives of all companions involved so that they will serve the purposes he wants them to, all while he sits in his tackily decorated room and goes "Dance, puppets, dance!".
Oh, and once that happens, it turns out that manipulating people's pasts to make them do what you want doesn't make people very kindly inclined towards you, and that part of Fei Wang Reeds gambit backfires too as Kurogane and Fai deal the finishing blow.
In Fairy Tail the Phantom Lord's Guild Matser, Jose Porla uses his super weapon Jupiter on the Fairy Tail Guild knowing Erza would most likely block the attack. This would leave the Fairy Tail Guild weak enough so that he could capture Lucy.
A pretty brilliant example from Kill la Kill, courtesy of Satsuki Kiryuin. Fed up of club presidents springing up left, right and centre, hoping to defeat her and gain Three-Star status, Ryuko forms a Fight Club to systematically take them all on, with Mako as club president so she'll get the privileges that come with the position. Much to the Elite Four's surprise, Satsuki approves the club's formation... because she knows that once Mako's family have had a taste of luxury, they'll want more and more, constantly pushing Ryuko to fight more club presidents to further improve their reputation and situation. Ryuko grows tired and, with the family constantly going off and doing their own thing, lonely...again, just as Satsuki predicted. Eventually, Ryuko disbands the club, but not without having to fight a greed-ridden, Two-Star-Uniform-wearing Mako, and snap her out of it using The Power of Friendship. To top it all off, it even combines with Xanatos Gambit; since the Fight Club pretty much wiped out all the "unworthy" club presidents, Satsuki can now elect new, better ones in their places.
Lupin Family All Stars: The whole trap is one on behalf of Zenigata and the gang falls for it perfectly. They'd've been fine if they had just ignored the suspicious invitations. So it was silly of him really to not only let Lupin get re-dressed, but also not search him for explosive contraband on an island made of explosives...