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.
Ensemble Darkhorse: 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.
People have taken to calling Araragi's mother Mamaragi.
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.
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/.
Magnificent Bastard: As said by Koyomi in Episode 8 referring to Oshino: "You... Even compared to a devil, you're such a magnificent bastard."
Moe: 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.
Nightmare Fuel: See No-Holds-Barred Beatdown on the main page; Suruga's constant hysterical screaming of "HATE HATE HATE HATE HATE HATE!!!" while savagely beating Koyomi is quite disturbing.
Tear Jerker - The last episode. When Araragi realizes that when Oshino told him "I'm going to leave this city one day", that in fact was his way of saying goodbye, because he doesn't like formally saying goodbye. Whilst we already know Oshino is a Jerk with a Heart of Gold, this shows that he is incredibly soft when it comes to goodbyes.