Series protagonist Shinji Ikari, the pilot of the Evangelion Unit 01, gets his Berserk Button pushed more often, and to more disastrous effect, than any other character. An ironic trait, as his usual demeanor is so timid and passive. Many of the key Berserk Button moments are involuntary, the result of Shinji's Eva unit being knocked out of commission... and a certain subdued consciousness (presumably his mother Yui) within the Eva taking control to avoid obliteration. On another level, however, the Eva units are adult extrapolations of the children who pilot them. Shinji's Berserk Button is often connected to his feelings about his father and about his own powerlessness.
Asuka Langley Soryu (or Shikinami in 2.0), the pilot of the crimson-colored Evangelion Unit-02, has many Berserk Buttons, all of them rooted in her deep insecurities. Usually questioning her skills or her robot is a very bad idea. Too bad it didn't come into play (until too late) when it would have counted in The End of Evangelion.
Although Misato Katsuragi's Berserk Buttons are generally tame compared with those of most of the cast, you still don't want to make any comments about her recent weight gain, her status as Christmas Cake, her generally slobbish lifestyle, or her deeper feelings for either Kaji or Shinji.
Dr. Ritsuko Akagi displays one hell of a Berserk Button late in the series after it's revealed to her that her beloved Gendou was merely using her for his own devices. In a fit of jealous rage, Ritsuko destroys all the remaining Rei Ayanami backup clones, meanwhile exposing many of NERV's secrets to Misato and Shinji. Later, she makes an ill-fated attempt on Gendou's life. Ritsuko had fallen into the same trap as her mother, and with the same man.
Does she make an attempt on Gendou's life, or does she just try to get the Magi to self-destruct? Either way, it fails, which leads to her prompt execution by Gendou.
It seems to run in the family. It's best for your own safety not to call Naoko Akagi an old hag - as Rei I discovered.
New pilot Mari Illustrious Makinami, introduced in the Evangelion 2.0 retelling Takes control of the Unit-02 in the fight against Zeruel, when Asuka is out of commission due to taking Touji's place as the Unit-03's test pilot in an alternate universe twist on the original story. Although very little about Mari is revealed as of the second film, it's shown that she's an incredible pilot. So good in fact, that she's able to actually activate Unit-02's long unseen Berserk mode with a simple phrase...in a sense literally pressing its Berserk Button undoing its Restraining Bolt. This has led many to consider Mari something of a possible Mary Sue, but since little about the character is known as of now, judgments are in reserve.
Minor correction: the "Beast" mode that Mari activates is actually the result of an upgrade, not a "classic" Berserk moment. That is why she could activate it by simply uttering a secret (not to creative) codeword: "Za Beasto!"
The Evangelion "robots" (In actuality, a very loose description) themselves, perhaps more than Shinji himself, could be considered characters with the real Berserk Buttons.