These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Crosses the Line Twice: Koyomi getting into a fight with a 5th-grader, beating her up (mildly), and then laughing about it. The fact that she's already dead may or may not make this worse. The light novel's description of the fight made it worse, detailing liquids that came out of her mouth that may or may have not been stomach acid.
Shinobu: Fifth place on the popularity polls while not speaking a single word and getting relatively little screen time is pretty impressive.
Nadeko: Recent character popularity polls has confirmed that she is currently the most popular character of this show, despite her arc is the shortest among other sub-heroines.
And now Kaiki is one too.
Everyone Is Jesus in Purgatory: Fans have theorized that all the oddities simply stand in for otherwise mundane life problems; the ones with the most merit are Hitagi's (her "weightlessness" being caused by an eating disorder), Suruga's (well, extreme jealousy, that's barely a stretch there) and Hanekawa's (her problem could disassociative, multiple personality, or bipolar disorder, depending on the interpretation of Black Hanekawa and the Tiger; it might even just be normal lashing out from stress). Nadeko would be "constricted" by the thoughts and actions of the jealous and spiteful, and Mayoi could be the Adult Fear of a child lost forever.
Fashion-Victim Villain: Deishuu Kaiki. A tie you could jump rope with, that one. And that dress shirt is damn near an actual dress.
Fetish Retardant: When the Meddlesome Cat grabs Koyomi from behind, he licks his neck in a very sensual way. What's the problem? That the Cat's tongue is sharp enough to open Koyomi's carotid artery.
Foe Yay: Senjougahara and Keiki. Keiki believes that when they first met, Senjougahara fell in love with him (which she denies). The Hitagi End arc revealed that he admired and cared about Senjougahara a lot (he destroyed the cult her mother joined and arranged for the divorce, believing it was best for her). His irritation with her boyfriend and her proposition of prostituting herself implies that it's not a platonic love either. Senjougahara admits herself that she came to enjoy their time together and even flirts with him a little (joking about how he must have been excited to have a high school girl at his command and the color of her underwear).
Harsher in Hindsight: When Tsubasa finds Koyomi beating up Mayoi, she tells him that it's wrong to hit a child without telling them a justifiable reason. We later find out that Tsubasa's stepfather physically abuses her. It's likely a deliberate example; Tsubasa has Scary Shiny Glasses as she makes the former statement.
Internet Backdraft: Thanks to an anonymous user now called "Gorespammer" relentlessly trolling and reverse-trolling the series, it's pretty much impossible to have a serious discussion about it on /a/.
Les Yay: Between Senjougahara and Hanekawa during the Tsubasa Tiger arc. The cherry on top was when Senjougahara insisted that they shower together (which Hanekawa agrees to).
Magnificent Bastard: As said by Koyomi in Episode 8 referring to Oshino: "You... Even compared to a devil, you're such a magnificent bastard."
Well, yeah. Though not really a harem anime, it has one guy and a bunch of girls, all of whom fall into traditional moe characterizations. In fact, some of the more Genre Savvy characters make fun of it, like Oshino's habit of calling Senjogahara Tsundere.
Ironically enough, Nadeko, who's arguably the one who fits the actual definition of moe the most, turns out to not be moe at all.
Okay, Araragi, it's pretty despicable that Hanekawa's guardians don't give her a bedroom and force her to live in the kitchen and hallways, but this is hardly the kind of revelation that should prompt panicked hysterics...
Senjougahara's expression when she finds Hanekawa in the first episode of Monogatari Second Season. They probably were going for showing how worried Senjougahara was through an unusual amount of expression. What they made was just silly. Also while the slap was decently animated, Senjougahara's followup by pounding her fists on Hanekawa loses a lot of the intended impact because she looks like her fists are moving through molasses.