The Artifact: Leon, who popularity has allowed him to continue staying with Larry long after any story point justifies his existence. After the rest of his family leaves, Leon ignores Larry's hints and continues to freeload off of Larry. Only in one later episode does Larry finally try to get rid of Leon, but his ploy backfires.
Darkness-Induced Audience Apathy: The Kafka Komedy that was normal in Seinfeld is turned Up to Eleven on this show. While you can easily say that Larry is enough of a jackass that he deserves plenty of the punishment that comes his way (especially because he doesn't really knows when to back down or tone down the asshole), there are also many examples where the universe just seems to have it against Larry for no reason other than him existing (let alone just being in the wrong place at the wrong time), and this (plus the typical cast-ful of belligerent assholes that is normal to a Larry David production, including Larry himself, and the terrible things they do) can make it hard to find the fun in the gags, no matter how good the set-up. (And if you want good In-Universe examples of "Dude, Not Funny!" to illustrate this... well, look two Tropes down).
Ear Worm: The entire goddamn soundtrack. Special mention goes to "Frolic" (the show's theme).
Life Imitates Art: The moment in "The Lefty Call" where Larry tries to call out a skinhead, only to find out the man's actually a cancer patient, basically happened in real life.
Mary Sue: Subverted to hell and back. Larry is a deeply flawed character who people, quite rightfully, cannot stand most of the time. While there are occasions that he is sympathetic, he commits acts which are pretty much inexcusable from any viewpoint (he once nearly let Jeff and Susie's daughter drown because he didn't want to damage his blackberry and deceived his cancer-stricken girlfriend into leaving him).
Never Live It Down: Bill Buckner in-story. 25 years after committing a mistake that cost the Red Sox the World Seriesnote (actually cost them Game 6, tying the series and necessitating a Game 7, which Boston lost), everybody still hates him for it.
Susie Greene. That is, if you don't find her hilarious. But just imagine Jeff's life with her...
Martine, the nanny from hell, is quite unnerving and creepy. For starters, she's overly giggly, which is already unsettling on its own. And according to Cheryl, she's full of non-sequiturs, which is also a little weird. But she becomes downright crazy when she hears the Looney Tunes theme (due to hearing that theme over and over again at the Looney Tunes Lodge, where she worked for 15 years). She also hums it over and over and over. You know she's terrifying when even Susie thinks she's a mental case.
Squick: Larry and company talking about a child's massive penis like it's a normal thing to talk about can be really uncomfortable to watch.
Strawman Has a Point: Larry is almost always the object of scorn and derision for his actions and opinions, but a fair bit of the time the audience is meant to sympathize with his refusal to go along with various abritrary and capricious rules of "polite" society, believing that Larry is typically in the 'right'.
Tearjerker: When Sammie chooses Oscar over her dad. That's gotta be devastating to Jeff.
They Wasted a Perfectly Good Plot: While the eighth season was touted as the "Larry in NYC" season, only five of the ten episodes feature this setting, with the rest taking place in L.A. as usual.