Akihisa and Yuujithrowing the test papers from the roof of the school, forcing the teachers to prepare a new test and nullifying Nemoto's attempt at cheating.
In Season 1 Episode 11, Akihisa, Yuuji, Kouta, and Iron Man each have a CMOA. Yuuji's is for intentionally letting his grades fall until the very last test to throw off their opponents for the ESB, then letting the poor grades become known to their opponents so he could surprise them by overpowering Class D's summoned beings. Akihisa had his avatar punch right through a wall dividing the classrooms, even though his own bones began to crack and his knuckles bled, but even that turned out to be a decoy for Kouta and Iron Man to get their CMOA, as Kouta crashes through the window with Iron Man to set up a surprise battle in health to finish off the Class B rep.
Nemoto: He can't do that! He can't jump in there with the teacher! Fukuhara: Oh, but he can. Iron Man is not only able to approve battles in every subject, but he can jump with the best of them.
Yuuji gets an awesome one defending Shouko after a couple insults her publicly about her dream to be a bride in Episode 7 of Season 1.
Class F almost defeating Class A's superior academic strength with clever tactics near the end of the first season. They could've easily won the battle had Akihisa's Iron Bracelet not died at a crucial moment.
Akihisa had a heartwarming moment with Minami during Episode 8 of Season 2. On Akihisa and Yuuji's side however it becomes a Moment of Awesome, as told on the light novels. When Minami's Classics textbook got messed up, Akihisa and Yuuji did their best to replace it, including riding downhill with both of them in the same bike just to chase the container van which had the school's extra supply of books taken away. Counts as Establishing Character Moment for both Akihisa and Yuuji's... uhh... weird friendship.
Akihisa doing a Big Damn Heroes moment in Volume 9 of the light novels. He went in late due to a... uhh... certain incident, but by the time he arrived, 2-F was already cornered into submission on their war with 2-C. With the help of 2-A, he got behind enemy lines to cut through to where Mizuki was, who was about to be victimized by 2-C and 2-B's plan to disqualify her from the ongoing war, which will cripple class 2-F's chances of winning. He gets there on time, completely destroys their enemies' elaborate plan, which also involved a subverted Red Herring war against 2-A and 2-B, and Mizuki and Yuuji proceeded to finish the summoning war via a Curb-Stomp Battle.
Of course, Akihisa and Yuuji's final battle of the summer OVA competition counts. After purposely getting hurt, Akihisa turns a losing battle around by letting his opponent block his teammate's shot for him. This leads to that guy getting knocked out and Akihisa avatar unleashing a Limit Break that warps the battlefield and deliver a No-Holds-Barred Beatdown to the last guy.
Just about everything Iron Man does in the anime, ranging from appearing from out of nowhere to drag unfortunate students off for remedial classes, to the aforementioned window crush and approvaling every subject for a battle, and even saving three students when the roof collapsed by hoisting up them all up to safety in one go.