The first real fight, the one between Stiyl and Touma, is pretty cool in the anime and manga. But in the light novels it's narrated from Stiyl's point of view, and we see how a magician views Touma and why he becomes The Dreaded. At one point, Stiyl seriously thinks he might be fighting God.
The ending of the fight where Touma destroys Innocentius with an Offhand Backhand before running up and punching Stiyl out.
When Touma and Stiyl face off against Aureolus Izzard, a Reality Warper. After immobilizing and toying with them, he instantly completely removes all of Stiyl's skin and severs Touma's right arm, the source of his power. Touma cuts off Aureolus's subsequent Evil Laughwith an Evil Laugh of his own, walking steadily towards him—blood squirting out of his stump—with an insane smirk on his face and mocking Aureolus's frantic attacks as they pass right through him or are otherwise negated. Turns out that Aureolus's power was controlled by his thoughts, and Touma's insane behavior so terrified Aureolus that he started believing that his incredible powers had been rendered completely ineffective. By the time Touma reaches him, his imagination has granted Touma a huge dragon head in the place of his missing limb. Even after seeing his companion's skin ripped off and getting his special arm chopped off, Touma was still able to bluff his way to victory with a Slasher Smile on his face.
Before that, the 13 Knights of the Roman Catholic Church blow up Misawa Cram School... only for Aureolus to literally reverse the damage and annihilate the Knights.
The reality warping technique, Ars Magna, was thought to be impossible because one needs to chant for 100-200 years to get it. Aureolus cut that time drastically by brainwashing everybody in Misawa Cram School to chant for him.
Misaka Imouto using her powers to make ozone in an attempt to suffocate Accelerator. He saw through the tactic and attacked her, but he complimented her on finally figuring out a way to hurt him.
Which bares the Fridge Logic that if she was battling Accelerator..why not spray the room with tasteless, oderless knock-out gas?
During Touma's fight with Accelerator when he starts making plasma, Sister's make good use of their powers by making wind turbines turn dissolving Accelerator's plasma in the process. Complete with Theme Music Power-Up!
Touma's fight with the Unabara impostor in the construction site.
Touma blocking the giant stone golem Ellis' punch with one hand.
Index herself gets one when, after discovering her Sherry Cromwell decides to use her golem to attack Index and get the 103,000 grimores that she has memorized Index then shows even without going into her emergency survival mode she has complete control over those grimores and uses them to show that the giant golem she is fighting and that Sherry's abilities aren't even on her level as she proceeds to methodically destroy the golem all while the song Jellyfish plays in the background.
Touma's Big Damn Heroes moment in episode 24 of Index, when he comes out of no where and defeats the enemy with a single punch, all while reassuring a friend. Nice.
What about Hyouka who pulls of her own Big Damn Heroes. As Index is about to get squashed by the golem she jumps at it and not only topples it but even caves in its chest. And then proceeds to stop it's fist and even pushes it back.
Touma saving Kuroko by leaping up using furniture thrown up by Mikoto was epic. Finishing it by punching a partially teleported object backwards through dimensions. There are no words.
Kuroko gets one for the fight prior to that happening. She was severely outclassed by another teleporter in all categories (including being able to teleport objects without touching them) except for speed. Had she not been wounded by her opponent's surprise attack the fight before, it's entirely possible she could have won. Even when she's pinned down she delivers a "The Reason You Suck" Speech, causing her foe to drop over 4000kg of weight on Kuroko in rage.
Immediately after above scene, Awaki was seen escaping Cue Accelerator, of all people, showing up to wreck Awaki in the face.
Touma: What should I do? WHAT NONSENSE, OF COURSE I SHOULD SAVE HER!
Touma knocking Biagio Busoni's giant cross away with his left hand.
Accelerator, pissed off after a call from Kihara Amata who tells him that he holds Last Order captive and threatens to kill her, activates his remaining esper power, shoves his arm in the wall of the building besides him and THEN PROCEEDS TO THROW THE WHOLE BUILDING AGAINST ALEISTER'S FORTRESS. Just wow...
Aleister: Seems like there's some commotion outside...
In the novels, although he doesn't throw a building, he borrows a huge amount of energy from the planet's rotational vectors to fuel his attack - enough to slow it down by 5 minutes.
And as a testament to how invulnerable Aleister's fortress is, the building remains unharmed. To hammer in the point, Accelerator's attack used enough energy to slow down the Earth's rotation for five minutes! (For reference, that amount of kinetic energy would be more than enough to shatter planet Earth several times over.)
The Academy City Invasion arc as a whole can be considered a single Crowning Moment of Awesome for anyone involved. Especially for Accelerator.
Accelerator outwitting the leader of Skill-Out and defeating him. Touma vs Hamazura, a few minutes after Touma had been shot in the side.
Mr. Misaka's phone call in the end of SS2, asking someone at Academy City why girls who look exactly like his daughter have been attacking military and research targets (in an attempt to rescue the gemstones). He makes remarks about what he'd do to anyone hurting his daughter, and then the truly awesome part hits: He's talking directly to Aleister. Yes, that's right. A wandering consultant is threatening possibly the most powerful man on the planet.
During the Gemstone incident, Academy City cannot deploy fast enough to rescue them from the CIA or rely on their cooperative institutions, leaving them out of options. Well, mostly out of options.
Kumokawa Seria:I don’t like owing people favors, but we should just bow down and ask that frog to help us.
“Galashiels, UK, Higher Contact Society. Misaka has suppressed the fourth, eighth, and thirteenth domes,' reports Misaka #17000.” “Lausanne, Switzerland, World Intellectual Club. Misaka has taken out an armed guard,' says Misaka #18022 as she provides her confirmation.” “Guadalajara, Mexico, Sixth Sense Headquarters. Misaka has blown up the main entrance to the research building,' says Misaka #14333 as she heads further in.” “Deseado, Argentina, Human Sports Analysis Center. Misaka has finished taking control of the secure electronic lock,' says Misaka #15110 as she opens the route to the secret area.” “Davao, Philippines, Humanity’s Wisdom Headquarters. Misaka discovered an escape boat and is going to take care of it beforehand,' says Misaka #10090 as she begins her attack.”
Give one to Uiharu for managing not to break under Kakine's torture just before the above event, including putting her life on the line for Last Order.
Volume 16: Touma showing up to saveKanzaki and the rest of the Amakusa Catholics from Acqua of the Back when he is already heavily injured from a fight with the same person. Not only that but this gave the Amakusa Church a chance to take down Acqua with a Heroic RROD-inducing spell.
Touma proves in Volume 17 that despite his label as one of the idiot trio, he's much smarter than he lets on. He pulls one over a terrorist by pouring boiling tea into a narrow air vent and causing it to expand, making the terrorist on the other side think that there's someone inside the air vents. The terrorist decides to check the air vent... and gets boiling hot tea poured on his face. This, by the way, was meant to distract the terrorist so Touma could enter the cargo hold through the door safely. Also, he ends the fight by giving the terrorist a lesson on thermodynamics.
Touma: “Hey, terrorist. Do you know about thermal expansion?”
Volume 18: After Fiamma steals Index's remote control, Touma simply says: “Stiyl,” Kamijou called out to the red-haired magician. “I’m gonna go punch Fiamma. While I’m gone, you take care of Index.” BADASS!
Volume 21: The combined efforts of Accelerator, Hyouka Kazakiri, and Acqua of the Back weaken and drive off Archangel Gabriel.
Volume 22: Touma drops the "Star of Bethlehem" on an Archangel. And because this was not enough, he runs through the entire thing at the same time, so HE COULD PUNCH THE ARCHANGEL PERSONALLY!
Also, when Fiamma of the Right severs Touma's right arm and absorbs it, becoming La Persona Superiore a Dio aka The Person Higher Than God. Fiamma gloats and attempts to kill Touma, but "The Invisible Thing" emerges from his stump, terrifying Fiamma. The dragon attempts to kill him, but Touma declares that he doesn't need its help and absorbs him, regenerating his arm. He proceeds to talk to Fiamma, making him doubt whether his plan was actually worth it, then charges up and punches him out.
As Fiamma is being defeated, he takes comfort in knowing he will win in the end, as a massive wave of telesma will fall on and annihilate Earth. Accelerator Awakens completely, turns into a shining angel, and reflects the wave back into space.
This might have more repercussions than you would think. By New Testament 2, Aleister is backed into a corner so deep, he took NO actions out of fear of disrupting his plans further, even when AC was about to be destroyed.
In the Short Story, Uiharu SS, Uiharu jumps atop a vehicle that's moving at 120 kph, then ties herself to it and uses a parachute to turn herself into a human brake!
In the Gemstone SS, Uiharu winds up in a hacking confrontation with Mikoto. Uiharu wins! Admittedly, by effectively destroying the data Mikoto was trying to steal, but still.
New Testament 2: Birdway got away with screwing over Aleister, and pretty much ensuring that Touma, Accelerator, and Shiage will involve themselves more in the next battle using her skills at analyzing charismatic figures. No wonder she's the author's favorite.
In New Testament 3, she's even more Badass. The other heroes are facing a magician who's basically Cinderella and curbstomping everyone, when she simply hits her with a Tanker truck and then blew it up.
Also from NT 3: Touma and Umidori defeat Saronia A. Irivika even though she paralyzed the left side of Touma's body.
New Testament 4: One-Eyed Othinus curbstomps everyone, even Touma. She severs his right arm, but when "The Invisible Thing" emerges from the stump, she curbstomps it too!!! Then, Ollerus and Fiamma of the Right arrive and manage to fight her to a draw and make her retreat.
Touma storms Baggage City and takes on Marian Slingeneyer. When Marian tries to use Dáinsleif to scare Touma to death, Touma is completely unafraid. When Marian threatens to draw Dáinsleif and trigger Ragnarok, Touma confidently points out that if she had the guts to do that, she would have done it already. She doesn't have the guts, and Touma punches her out.
New Testament 5: After being just about everyone's Unwitting Pawn since the very start of this series, Touma has finally had enough. Working with Reliable Traitor Thor to screw over all the sides this time, as well as showing off some impressive skill with The Plan such as tricking both Gremlin and Ollerus' group, specifically freakin Birdway herself, into thinking the other is setting up a trap. Looks like he's growing into quite a Magnificent Bastard isn't he?
The ending. An overzealous Anti-Skill member mistakes Touma for a bad guy and shoots him several times. He is rushed to the hospital and treated by Heaven Canceller, as usual. The doctor discusses with some people about how Touma will be handed over to Anti-Skill when he recovers. Touma, who was conscious the whole time, escapes from the hospital despite still being in serious pain.
New Testament 6: The whole novel is a huge one for Mugino, who gets past all her fears and doubts, allowing her to curbstomp a guy pulling a Shape Shifter Guilt Trip on her. She declares that she loves Shiage and that love has made her stronger than ever. Then, she teams up with Accelerator to take on Kakine. The fight ends with Beetle 05 declaring that he is Kakine, taking over all of Kakine's Dark Matter creations, and erasing the original from existence.
Touma, still weakened from having been shot several times, fights back-to-back battles against Leivinia and Thor, even breaking Thor's wrists. Then, Touma figures out how to cure Fraulein's Horror Hunger.
4. The battle between the Level 5s and the 7500 "heroes"... Or better said, the Curb-Stomp Battle. Accelerator alone defeats 1000 just with his Dynamic Entry, Gunha is more concerned about preventing the forest fire, Mugino blows up a truck and then makes small talk with ITEM, and Misaki and Mikoto are spending more time bitching to each other. Oh, and that person who everyone thinks is the #6? Actually, just some Badass Normal whose punches are described as "like machine pistons"!
5. Kakine showing his new merit as Big Damn Hero, saving the day and fighting side at side with Touma against Rensa #29.
6. Fremea finally deciding to stand up for herself instead of waiting for a hero to save her and hacking the city's AIM field despite being a mere Level 0.
New Testament 8:
Loki anticipated a spy would infiltrate GREMLIN's base, so he placed an artificial sky over it so that Ollerus would read the wrong stars and give his allies the wrong directions.
Touma and Freyja's Traintop Battle. When she sends her army of monsters at him, he negates one, crushes one underfoot, then tackles one so hard it flies into the others and knocks them all off the train. Later, when Freyja thinks she's knocked Touma off the train, he reveals he hung on, then throws his jacket over her head to blind her.
Mikoto lands gently from a plane sliced in half while catching Index, leaps from building to building too fast for an interception spell to keep up, and blasts Níðhöggr to pieces with a volley of coins. She even has the attention to spare to feel bad about upsetting Sphinx with her magnetism.