During a Defense Against the Dark Arts lecture, Quirrel tells Harry that he must shoot somebody with a hex, or else he would deduct house points. Rather than prove Quirrel right that Harry has a killing instinct, Harry shoots himself.
In Chapter 10, the raised by rationalist/scientist Harry meets the first challenge that is able to seriously oppose him mentally, the Sorting Hat. Why? Because the Sorting Hat runs off the consciousness of the person who wears it, meaning that the Hat has all of his smarts but retains its own goals. The resultant battle of wits has to be seen to be believed, being both tense, intelligent, and hilarious. The crowning moment of the scene is the Hat's last laugh, where even after failing to convince Harry to join Hufflepuff or Gryffindor, briefly declares him Slytherin before saying "kidding, Ravenclaw". This finally gets through to Harry the danger he incurs by relying excessively on his intellect without any consideration to his or other's wellbeings, and ensures that his stay in Ravenclaw will be properly tempered.
When Neville summons Harry to scare off some Gryffindor bullies — who bullied the son of Bellatrix Lestrange, the woman who tortured his parents.
Chapter 28. Harry manages to rewrite an immutable law of Transfiguration.
Hermione gloating at the end of the first battle of the armies.
"I can do anything if I study hard enough."
Also, in Chapter 32: Either Harry Potter had thought of a lot of very good ideas very fast, or for some unimaginable reason he'd already spent a lot of time working out how to fight underwater. Yes, completely unimaginable.
Neville's complete and utter Ron pwnage.
Ron: Curse you, Longbottom! Can't you ever fight without your dumb special attacks!? Neville: I don't fight fair! I fight like Harry Potter!
When Draco finally managed to land a hit on Neville, everybody nearby hit him again just to be sure he stayed down.
Chapter 34: Professor Quirrell chewing out the wizards for having to be saved by a Deus ex Machina. From a Dark Lord who commanded only fifty followers, no less. Not that his proposed solution would have been a good thing...
Chapter 39: Dumbledore accuses Harry of wanting the secret of the Dark Lord's immortality in order to use it for himself. Harry's response: "Wrong! I want the secret of the Dark Lord's immortality in order to use it for everyone!"
Chapter 45: Harry kills a dementor. No, he does not drive it back, he KILLS IT, by completely rejecting what it symbolizes. His patronus is a Homo sapiens sapiens, BTW.Cliffnotes of the awesome.
For every person that thought "boring cliche golden patronus, this sucks" upon reading this, give it a chance. It's AWESOME. It's awesomely justified, fits absolutely perfectly with the explanation given for dementors, and simply rocks.
Merely a Homo sapiens sapiens? This troper likes to think that Harry's patronus was the God Emperor himself.
Don't forget that his Heroic Willpower is so strong that he eventually learns to ignore the dementor's negative emotions by himself — although he has a tough time telling the difference between artificially-induced self-doubt and the rational realization that something is a bad idea.
"And they continued their journey downward, as the Dark Lord reached into his pouch, and retrieved and ate a cookie."
What with the author's confessed fondness for a Certain Japanese Franchise. This troper prefers to read this as: the Dark Lord took a cookie. AND HE ATE IT!
Chapter 58: This. Is. My. Broomstick!
Goddamn lightsaber battle. That is all.
Indeed. Daphne Greengrass vs. Neville Longbottom OF CHAOS. I think we can all agree that they BOTH Took a Level in Badass.
Hermione notes that it's a good deal less impressive if you've seen certain movies, but it was still very impressive.
Tracy channeling Harry Potter's powers and gluing a large group of bullies to the ceiling. It's safe to say that anyone connected to Harry Potter should not be messed with.
Using a ritual that, if it was an actual ritual, would have involved permanently sacrificing the Outer God Yog-Sothoth to briefly maintain Harry's presence, no less.
Chapter 78. The Dragon Army literally defeats the Chaos Legion at their own game. How? "NOW!" bellowed Dean Thomas... "DO WHATEVER THEY DO!"
In chapter 81, Harry forces Lucius' hand by calling in his blood debt for killing Voldemort, accepts the extra 100,000 galleon fee without hesitation, swears to destroy Azkaban (which makes Dumbledore back down and enlightens the Wizengamot as to who they're dealing with), and forces a Dementor to flee into a corner without casting a Patronus Charm.
(to Umbridge) "I make you this one offer," said the Boy-Who-Lived. "I never learn that you've been interfering with me or any of mine. And you never find out why the unkillable soul-eating monster is scared of me. Now sit down and shut up."
Chapter 86. Harry Potter vs Alastor Moody.
"Really, boy?" said Mad-Eye Moody, his blue eye spinning rapidly. "I'd say that's a little... paranoid...
NOT PARANOID ENOUGH!"
From the same chapter: Harry Stuns Mad-Eye Moody in a duel.
Chapter 89: Harry kills a troll (albeit one down an arm) with only a levitation charm and Finite Incantatem. Well, and a Transfiguration spell to turn part of its brain to acid.
The foundation of the Auxiliary Protective Special Committee in Chapter 98 is a crowning moment for each of the students and their sponsors, not just Harry. In fact, it might be Draco's first real CMoA.