Alternative Character Interpretation: If Harry's theory about how Fred and George pranked Rita Skeeter is correct then the lovable tricksters could be seen as mind-raping someone with a false memory charm to destroy their career, thereby exposing them to the ire of very dangerous people. Harsher in Hindsight much?
There's a long running debate within the fandom on whether Harry is a sociopath or not.
Once Harry realizes all the utility of a portable time machine.
God he loved his Time-Turner and someday, when he was old enough, they would get married.
Harry was in love. It would be a three-way wedding: him, the Time-Turner, and Professor Quirrell.
Crack Pairing: Sirius Black/Peter Pettigrew. Unsurprisingly, it apparently didn't end well.
Critical Research Failure: The Stanford Prison Experiment is taken at face value and treated as a serious piece of research; in real life it's considered a complete joke which functions as a checklist of how not to do research (as explained on its own page).
Lampshaded when Daphne Greengrass says Lucius Malfoy would skin his son alive and turn him into trousers if he heard him talking against blood purism.
"Draco Malfoy smiled, metallic robes gleaming in the light of his full corporeal Patronus; it was a smile both arrogant and dangerous, like being turned into a pair of leather pants was beneath his concerns."
Evil Is Cool: He makes it look so and Harry admits his arguments are very convincing and tempting.
Chapter 17, in which Harry attempts to factor the product of two prime numbers using the Time Turner? (More generally, he comes up with a method to solve NP-complete problems, that is, problems where you can check a solution quickly, but finding that solution requires searching a gigantic solution space.)
Chapter 4, with its off-hand mention of how someone could make a lot of money by exploiting the difference in the relative prices of silver and gold in the Muggle and Wizarding economies. (Buy gold from wizards, sell it to Muggles, use the money to buy silver from Muggles, sell the silver to wizards, and end up with more money than you started with. Or, if the imbalance is in the other direction, buy silver from wizards and sell it to Muggles for gold.)
In chapter 26, Harry and Quirrell have a discussion in a scrying-proof location called Mary's room.
The "Slytherin System" described in chapter 63 is a version of onion routing, the anonymous networking technique used by Tor.
For those paying attention, the realization that the Pioneer Plaque was turned into a horcrux is the reason that Quirrel is dying - he has separated out his life essence over such a large distance that light speed is becoming a real problem. This is obvious pretty early on for those who are familiar with relativity, and when he (much later) reads about Muggle physics, Quirrel realizes his mistake. That the starry field is also a view from around the horcrux - and thus, Pioneer, far out in space - is also implied, but never stated outright.
The magic mirror in this continuity doesn't show your Heart's Desire, it creates a reality based on your Coherent Extrapolated Volition, which is a term the author coined himself as an idea to make artificial intelligences safe.
Growing the Beard: As the (edited) first chapter puts it: "This fic is widely considered to have really hit its stride starting at around Chapter 5. If you still don't like it after Chapter 10, give up."
McGonagall: What sort of contingency do you imagine this kit might prepare you for, young man?
Harry: One of my classmates gets bitten by a horrible monster, and as I scrabble frantically in my mokeskin pouch for something that could help her, she looks at me sadly and with her last breath says, 'Why weren't you prepared?' And then she dies, and I know as her eyes close that she won't ever forgive me -
As of chapter 89, it turns out that kit didn't help Harry as much as he'd been hoping. Mainly due to the timing of his arrival.
Magnificent Bastard: Draco took lessons on how to be one, but so far he's only a beginner. The true Magnificent Bastard of this fanfic is, of course the one and only Professor Quirrell/Lord Voldemort. Without spoiling anything:
In the /r/HPMOR subreddit, there are a good many mini-memes, especially after the excitement of the final story arc being posted. A few more notable ones are "still-in-the-mirror" theories, Harry's "pouc", and the Cedrics Diggory.
The author, like a lot of readers, seems to have missed the point of Ron, and "sees no reason for him to exist". Canonically, Ron represents emotion, as Hermione represents reason. Largely excising him from the story leaves out some great opportunities for character development for the hyper-rational Harry. On the other hand, their friendship was founded largely on Harry not being kind of a jerk.
When Harry catches the Remembrall, and it glows like a sun.
McGonagall's speech about the dangers of transfiguration.
If you transfigure your hair blonde, it will fall out. If you transform your skin to be clear... you will be very ill and spend a long time at St.Mungos.
The terrifying Welcome to the Real WorldOmake in Chapter 64.
After meeting the Dementors, Harry briefly becomes hyper-rationalist. Specifically, he divides everyone into useful or non-useful categories. If they aren't useful, if they're doing so much as annoying him by crying, he wants to kill them.
Ron the Death Eater: Subverted. While early on, Harry is very dismissive of Ron and it looks like he's going to be written out of the story, in chapter 31, he shows up again, as one of two chief strategists in Hermione's army. Unlike Harry, Hermione is better at recognizing the talents of others, and Ron is a master chess player.
Second Verse Curse: Harry only knows the first and last couple of lines of Tom Lehrer's "Be Prepared"; his parents rather wisely didn't teach him the rest.
Tastes Like Diabetes: Harry's "romantic" first date, organized by the rest of the girls in his class. He's not amused.
Toy Ship: Harry and Hermione are eleven/twelve years old, even if they don't act like it. Everyone believes they will end up together - except the 'loving couple'. Harry especially complains that it's hugely disrespectful towards Hermione for people to reduce her to just his love interest when she's a genius and heroine in her own right.
The Untwist: Chapter 104 reveals that Quirrell is really Voldemort.
The Woobie: Poor, poor Bellatrix. Quite remarkable, considering that her characterization here is merely an expansion upon her characterization in canon.
Subverted, the part where she started out perfectly decent until Voldemort "ruined" her was a complete fabrication to manipulate Harry. She was apparently a deranged blood-purist zealot long before he ever came along.