If you make your Sims very Neat in their personalty, they will always make their bed when they wake up and you can't stop them once they have started.
Doubling as The Scrappies, the street musicians introduced in Unleashed (W. C. Friendly and Giuseppi Renni — the former appears on community lots with an espresso bar, the latter on ones without), due to their habit of blocking doors and stairs while performing and taking their sweet time to perform before moving out of the way, which often causes Sims who were about to leave the lot to have their motives severely depleted by the time they manage to go home, at least if the player doesn't just quit to neighborhood in frustration first.
Starting from Free Time, Mr. Humble will give you a new computer. Always nice - but he does this for every family you create. Needless to say, mods were instantly created to disable this since the very high end computer messes up "rags to riches" scenarios. Some are also annoyed by the Product Placement aspect of it (the computer includes a copy of the then-upcoming The Sims 3).
"It's raining!" "It's hailing!" "It's snowing!" Yes, it's raining and hailing... how is it possible to get the "It's raining!" off of your task queue?
Put a computer in the house and the sims will always try and play on it unless you have free will off. This gets very annoying when you try to make your sims stand outside so they don't miss the school bus or the carpool... and they decide to go play on the computer.
Put a stereo in, and you better get used to the music - because it's not turning off (unless you turn off Free Will in the options menu, so they won't try to turn the blasted thing on again).
Pregnancy causes hunger and energy to deplete so quickly that failure to juggle both adequately is quite likely to prove fatal unless you micromanage like crazy. (Giving birth is 100% safe in the Sims' world, but pregnancy cravings can be a literal killer.) Gets even worse with Nightlife installed, as you'll often find yourself in the extremely frustrating position of watching your pregnant Sim starve to death after she (or he) falls asleep face-down in a plate of food. Luckily, the pregnancy mechanic was overhauled comprehensively in The Sims 3 to prevent this from happening.
Hope you enjoy the gesturing in Bon Voyage, because once you install it, that's all they'll do. Unless you get a mod.
The Free Time expansion adds some cool new items and the hobby system into the game. The problem is, once you Sim gets some hobby enthusiasm, 90% of their wants will roll up to be hobby-related. Have a new baby? Trying to get up the career ladder? Making a best friend? No aspiration points from those, because your wants are all "talk about hobby," "read hobby page in the newspaper," "practice hobby," etc.
The nannies are notorious for being incompetent to a dangerous degree (as in, "set the kitchen on fire by being too busy force-feeding a non-hungry baby to look after the TV dinner she's cooking for a toddler even though toddlers can't eat those" dangerous) and very glitchy (refusing to leave after being dismissed, getting stuck on the lot or even multiplying), to the point they're seen as the scrappies of the NPC characters.
If you hired a maid, and you have a dishwasher and an empty coffee cup sitting around anywhere in the house, the maid can get trapped in a loop trying when they put it away, pulling another coffee cup out of nowhere, and then putting that in the dishwasher.
The Memory System. It's a return of the Memory System of The Sims 2 in which Sims get memories added to their book whenever something significant happens in their life. Instead, Sims get memories for almost everything they do. Went to buy something? Memory! Just unclogged a toilet? Memory! Passed by the library on your way to work? Memory! It doesn't help that you get an incredibly annoying pop-up saying that your Sim obtained a memory every time this happens. Fortunately the developers had a heart and allowed you to disable the pop-ups, but not the memories. But wait! It gets better! Inactive Sims also gain memories. And memories take up space in your save file. If you've noticed that your game is loading/saving slower than usual after installing Generations, it's probably the game trying to process all of your memories. It's not surprising that within a week or so of this new system, mods immediately popped up to stop the memories. With the Seasons patch, they finally allowed disabling memories.
The Romantic Reputation system, which causes spouses to declare Sims a cheater simply for interacting with another Sim. It gets quite juvenile in some situations. For example, did a Flirty Sim decide to bat an eye to your Sim? Your Sim's spouse will then call you a cheater and proceed to reject all social interactions with you and even break up with you afterwards.
Then there are celebrities and vampires, which quickly become an uncontrollable plague. Thankfully, Supernatural and its accompanying patch give you the option to prevent them from spawning.
By becoming a celebrity, even by accident, even with the celebrity spawning turned off, your sims are potentially subject to scandals that give them a negative moodlet. Oh, and they also damage their relationship with every sim the affected sim is acquainted with regardless of the absurdity of the claim. This can often lead to half the town hating your sims for no real reason. You can offset the loss by personally interacting with each individual sim to assuage the rumors, and to sue for slander. This is exacerbated by the fact that some of the "legitimate" scandal types make no contextual sense; for example, Woohooing with a member of "the Occult" (i.e. any kind of supernatural being) will shame you in the public eye even if half the population have some kind of supernatural affliction or another, or if your Sim is one themselves, or if both your Sim and their partner are.
There's also the Ice-Cream truck introduced in Pets, which will frequently park in front of your home lot and stay there for hours like a stalker. Also the Paparazzi from Late Night, though they won't bother you unless you're a celebrity.
The "room" meter almost always keeps the Sims away from obtaining a perfect mood, and even in a very well-designed house, they still have it half-red. Thankfully abolished in Sims 4; since "moodlettes" co-exist and are not mutually exclusive, a house in disrepair will contribute to discomfort moodlettes, but it's no longer a need to have a spacious house (and you can get positive moodlettes for environment through decorating rather than trying to expand).
Supernatural adds zombies which never stop coming and constantly attack your furniture. It is pretty much impossible to have a garden without freezing the lunar cycle (the closest option you have to turning it off entirely.) A simple trick to safely garden is to build a perimeter or fence, then add a fence door. Left-click the fence door, left-click to "Lock...", and choose "Everyone but me" or "Everyone but my household."
Seasons continues the downward spiral of Scrappy Mechanics, with the attraction system which has a Most Annoying Sound to go with the hottie's attraction towards your Sim. Said mechanic also adds the submechanic of your Sim receiving love letters and gifts from other Sims in their mail or calling them on dates. Couple that with the unbalanced romantic reputation system, and your actually faithful Sim can still be considered cheating. The gifts and letters also tend to come in unnecessarily large numbers. There are mods that completely disable this.
The mechanic whereupon a new neighborhood is automatically populated with expansion and patch content buildings (Showtime venues, consignment stores, etc) when created. These new buildings occupy empty lots, seldom (if ever) match the neighborhood theme and are often placed haphazardly, either facing away from the street or smack dab in the middle of a large lot. This can leave a simmer spending several minutes after loading a new neighborhood rooting out, deleting and then repurposing the resulting empty lots so their towns don't look ridiculous. There is no way to turn this feature off, unlike methods to circumvent other scrappy mechanics.
The babysitter. Good lord. Picture this; your sim is a single parent with a baby and has to go to work to earn money, so naturally you call for a babysitter. The babysitter will be there for a while, but then a pop-up will suddenly appear where she says that she needs to go because she has another client, 'in the middle of your working hours! Not wanting to have the child taken away by the social worker, you have to get home as fast as you can to take care of your child, interrupting your work day and earning less cash! To make matters worse, the babysitter might even have done nothing but stare at the baby for the short time s/he was there, thus also leaving your baby with a terrible mood by the time your sim makes it home. Being a single parent (or two working parents) makes it incredibly hard to NOT lose your baby to the social worker...
Also in Generations, teens cannot drive unless being taught by an adult in the household (before, the teen was automatically able to drive, even right after aging up from being a child). This can be annoying, as sometimes the teen and/or adult will not follow the other to the car, thus rendering the entire situation completely pointless.
Story Progression can cause problems. The idea behind story progression is that when you are not playing a family, they will continue to progress (i.e. go to work, get jobs, etc.). However, this can mess up a lot of things. Sims can get in relationships, get married and break up, among other things. Story progression has even been known to cause babies to disappear from NPC households.