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Hayate in Hayate the Combat Butler quickly achieves this in the first episode. When Nagi is kidnapped and is being driven away in a car going 60 kilometers an hour, Hayate catches up to them in seconds on a bicycle. Then, when trying to stop the kidnappers, Hayate is literally run over by the car, but that doesn't stop him from leaping back onto the roof and scaring the kidnappers into submission. It's only then he collapses from his wounds (He was hit by a freaking car, after all!). It only gets better from there.
For instance, taking on a gigantic mecha with a machine gun as big as he is half-way through the series, despite having been sold to someone else...
Or when Hayate races against KITT (or CHITT as he's called) on a highway... bought by Nagi... and wins. Though then again, he won because Michael used his boost on a corner that was waaaay to sharp.
Of course, the above leaves out the fact that Hayate is on a perfectly normal bicycle, pulling the very large and heavy trailer that Nagi is sitting in and he was winning anyway.
The best bit is not when he races the car and wins... it's when he races a train and it ends in a tie. Mind you, by the look of it, it was one of Japan's superfast trains (albeit not a maglev).
...or ending a duel against Wataru in just seconds by catching the sword between his fingers and casually tossing it aside.
Considering his next actions (in the manga at least) is pretending to have lost, it's more of a Crowning Moment of Funny than anything else.
Especially in view of how bad his acting was. Really, that had to be one of the most clumsily thrown fights in history.
Really, Hayate's first crowning moment came before any of those, when he socked Santa Claus in the face (the guy had it coming).
Then there's unlocking the powers of the "Ultimate Butler Outfit" and employing a Finishing Move against a giant snake. The fact that the episode was done in the art style of Tengen Toppa Gurren Lagann only adds to it.
Hinagiku shows off how awesome she can be when Gilbert challenges Hayate to a beach volleyball match, with 'a weak girl' of Hinagiku as a partner while he has a volleyball robot. She starts off failing, until insulted about her chest. In the end, she wins the match against an army of volleyball robots, knocks Gilbert out (again) and Hayate isn't even implied to have touched the ball once. And she's still able to knock Hayate out after running away.
The "The End of the World" arc in the manga: It ends with a ''very'' emotional swordfight between Hayate and "A-tan" in a Field of Blades. Both of them were just 6 years old at the time, but it wasn't played for laughs here!
Chapter 27, page 13 of the manga. He turns into Ciel-senpai. Black keys and everything.
Chapters 252-253: You know the stone? The one Hayate has to keep safe or Nagi loses her inheritance? The one that, while intact could mean Athena being swallowed up for good by her super-powered evil side? Nagi smashes it herself. No inheritance? That's fine. She's got Hayate.
Chapters 262-263: Hayate's big brother rescues little!Athena from her solitary imprisoment in a magical castle.
Not sure of the exact chapter but it was in the early Greece Arc just before the Beach Volleyball Match; Gilbert challenges Hayate at the end of the previous chapter only to show him soundly beaten face first in the sand, he then gets up and tries again only to be sent flying. The third time is awesome as Hayate's apparently had enough distractions for one day and goes "Demon-Mode Hayate" with his Glowing Eyes of Doom and holding Gilbert by the collar asking him how much more he has to kick his ass until he's satisfied. Even Maria notes that beneath his cute/girlish face he can be ruthless.
As if to show us that Hayate can be a stone cold Bad Ass; Nagi contemplates what a pain it would be to get harassed by people out for her fortune like that more often, Hayate not only doesn't actually mind but he only feels sorry for the doctors which have work overtime to patch them up when he's finished with them!
Ayumu's moment comes in chapter 290 when they've visited a professional mangaka so that Nagi can get her work looked at. Nagi realizes that her work is horrible and scatters the piece before running away. Ayumu snatches away the page he's managed to grab, and then asks if the mangaka would read a piece Nagi's more confident about when it's ready. Ayumu has met this girl a few months ago (and instantly rivalry grew), befriended her less than a month ago, and is now defending her hopes and dreams.
A moment in chapter 353 literally left this troper saying wow for 2 minutes it was so awesome. Hayate has finally caught the cold that everyone in the previous chapter caught and at the most inconvenient time because at that time, he had to do his exams in order to continue going to Hakuou. After carrying a girl for 1km to the hospital wing on the other side of the school, giving advice and working out financial problems for Saki and Wataru for five minutes, and giving Isumi a large amount of blood so that she could defeat a huge monster that was threatening to devour the world, Hayate, all worn out and pale barely makes it on time to the exam. To pass the time, Yukiji gets out her PSP but she screws around and ends up having it hit Hayate on the head. What does Hayate do? Without looking up, he grabs the PSP and SNAPS IT IN HALF much to Yukiji and Izumi's utter shock. He was so pissed off in the end he didn't care about dealing with a pointless problem like that!
In the anime version of the Hinagiku birthday arc, Hinagiku sings Cruel Angel'sThesis. Which she supposedly does not know.
Ayumu concocts a Training from Hell session for Nagi. She even pulls in a professional mangaka to review Nagi's work.
Ayumu: "If it was to prevent Hayate from getting taken away, I'd hire a pro baseball player for a casual baseball game!"
Hayate's duel against the 'ultimate state of the art robot butler'. After the butler goes rogue, Hayate deftly maneuvers him into a losing position, but the butler points out that if Hayate finishes him off, Hayate will be electrocuted. He then points out that, no matter what, Hayate will still be poor for the rest of his life. Hayate breaks out a Slasher Smile.
Narrator: And you kids out there, don't ever try this at home.