Loli
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"In other news, children are getting sexier...and that's pretty cooooool."
- – Tom Tucker, Family Guy
The term loli is used on 4chan to refer to girls who appear to be prepubescent, aka "petite". The word is derived from lolicon, a Japanese term which is itself an abbreviation of Lolita complex. Within the anime fandom the term loli is often used to refer to any girl that meets the aforementioned criteria, regardless of sexualisation or lack thereof. A well-known example of such a loli is Yotsuba, the mascot of 4chan.
Some people contend that using the term loli to describe a character always implies some level of sexual attraction on the part of the speaker, which is false to varying degrees. There are some who also claim that a character's given age can exempt them from loli status (for example, the character Konata from Lucky Star, who is 18 years old but is generall seen as a loli due her design of being short, petite and having a delicious flat chest). Loli is frequently used to also describe pubescent but still underaged girls which is, technically speaking, not correct but most people let it slide as there is no specific term for girls aged 13-17 in wide usage among the anime community (although there is one outside the weeb community: Jailbait). The fact that most artists can draw an adult or a child but many find making teenagers who don't look like either a small adult or a tall child kind of harder to do possibly contributes to this. Ultimately as far as American law is concerned, underaged is underaged whether you're fifteen or eight. Also under American law is that drawings of fictional characters aren't people, since they aren't real and all that (barring tracings). This does not apply in Canada where it's an excuse to be stalling assholes at customs, looking through every page of comics and artbooks to make sure they don't have any loli. The general rule of thumb is someone stops being a loli when they look mature enough to start being described as a woman.
Related to this is the slang otaku term "moe" (萌え, pronounced [mo.e]), which literally means "budding," and a pun on "burning" with passion or light or life. One of the possible origins is believed to be Moe, a loli from the anime 恐竜惑星 (Kyouryu Wakusei lit. Dinosaur Planet) which is rather obscure. This originally described a sort of protective or paternal/maternal instinct invoked onto certain characters. These days it's either used by people to describe girls that are adorable and cute to the point that you want to take care of them.
The rule 63 version of loli is shota, which is typically preferred by either women or extra, extra creepy men (or some very sad men who want to vicariously relive their adolescent fantasies of being ravaged by an older woman). Much like loli, the term shota is also often used to describe pubescent boys rather than just aged twelve and under kids. As with their female counterparts, the dividing line between a shota and a simply young looking man is based on whether you'd probably call the person a boy or a man.
How accepting /tg/ or 4chan as a whole has been of loli and shota has varied quite a bit over the years, though it's worth noting there's always been those who are really into them as well as those who think people into it probably belong in jail. Most of /tg/ falls somewhere between the spectrum with most just choosing to ignore it if they aren't into it. The fluctuations are often caused by the people who are into it migrating onto another image boards, predominately from /b/, /a/ or /c/ (though /jp/ occasionally gets in on the action as well). Though as /tg/ is a work safe board and as of late that has actually started being enforced you're going to be seeing far less loli and shota, at least not any that would warrant bans.
Overall as fetishes loli is certainly on the extreme skub side of things. Sure there are a lot of fetishes that are much more bizarre and esoteric in the "why would anyone think this is hot?" sense as anyone who's looked at /d/ can tell you, but none of them carry anywhere near as much baggage. (Well, aside from furries, but that may not count as furries are historically detested for overlapping and overdoing not directly furry fetishes (e.g. scat, underaged, vore, among others), rather than or in addition to the "bestiality" argument, as well as just plain existing.)
Loli should not (and, indeed, cannot) be confused with Shortstack, the fetish for an adult-figured woman who is much smaller than the typical human man, typically around the height of a child. Indeed, shortstacks often have exaggerated hips, buttocks and/or breasts, which makes them quite visually distinct. "Oppai Loli" is the arguably even-more degenerate offshoot of loli, as it takes a loli character and presents them with adult-sized (and usually exaggerated) breasts on their otherwise childish frame, much to the dismay of many fans of shortstacks.
Dungeons & Dragons
The Glantri Known World Gazetteer has very incomplete rules for a child spellcaster: When they cast a spell, they have to roll 4d4 and if the result is over their age, they suffer one of twelve catastrophes instead. If an NPC, they also gain a level for every 12 months of study to a max of half the parent's level. Nothing else about this is mentioned. It says this should be reserved for NPCs, but then immediately says it can be possible as a result of a PC's heir taking over after their death.
Eberron has the Lolipope, but her stats block has nothing to indicate her age.
d20 System
Various d20 games, but not the original, have had rules and ways to play as a child character which will inevitably be used to play as a loli.
d20 Modern
d20 Modern actually includes the ability to play as a child character in the core rules, tucked away in the same section covering playing an old one. This manifests as a -3 penalty to strength and constitution and a -1 penalty to everything else for those under 12 and no penalty for those 12-15. While playing a 12 year old loli seems tempting, they are literally unable to complete character creation using only the core book without a high dexterity since every starting occupation but one (which has high ability score requirements) has, often bizarre, minimum age requirements and taking one is mandatory. Splat would fix this by introducing new occupations without an age minimum, and Critical Locations (a maps supplement of all things) would explicitly offer an alternative to occupations with no or low minimum age.
Star Wars d20
Star Wars d20 and its successor Saga Edition would retain the above mentioned aging rules of d20 Modern, though there's no starting occupation to create problems. It's tweaked slightly by giving 12-15 (and its non-human equivalents) a -1 penalty to all ability scores (which also apply to those under 12 due to how the aging system works). Unlike its parent system, Star Wars RPGs actually encourage using these rules and state it's most likely a character that starts as a level 1 Jedi is underage, especially during certain eras.
Pathfinder
Pathfinder 1e has a number of rules that get tied up with the loli trope. The first is rules that let you play below the average adventurer's starting age, which are simultaneously utterly terrible (for requiring you play an NPC class) and OP (it briefly mentions, as a weakness, your ECL is actually lower for this so your Adept gets extra XP and eventually becomes higher level as a result). Another is the absolutely wonderful feat "Childlike" which lets Halflings pass themselves off as human children. The first book of Iron Gods includes rules for Androids that look like children but have adult minds. The Kitsune exclusive feat "Realistic Likeness" breaks many of the general rules of polymorph spells, which should allow impersonating a child.
There is also the class archetype "Magical Child" for the Vigilante class that makes your transformation into your superhero identity flashy, allows you to have a familiar but gives you a rather poor spell progression using the Unchained Summoner spell list. The rules don't say you have to be a child to take the archetype, but it is heavily implied.
Lastly, the iconic character for the Kineticist class is a young human girl named Yoon. Despite breaking the rules, Yoon is playable as a pre-gen in PFS games and actually one of the better first level pre-gens thanks to her high AC and HP (avoiding the issue of first level PCs going splat easily) and virtually nothing resisting fire at that level. If you want to make everyone at a public game awkward without actually doing anything of note yourself, play Yoon for credit toward a Dark Archives faction PC. Half the early scenarios for the faction have a side mission that a slutty baroness offers a faction member sexual favors for completing, and these offers are in player handout stuff so the GM can't change it on the fly.
Mutants & Masterminds
Mutants & Masterminds explicitly supports playing as a child, due to a long history of child superheroes like Captain Marvel (the real one), Power Pack, and countless sidekicks. This has no inherent stats effect, Superboy is still super strong, but some trends (high power, low on skills) are suggested and it will count as at least one complication.
BESM 4e
Playing children isn't directly supported with any specific rules but it does provide some guidelines, chiefly reducing the amount of Character Points a character has at character creation (it is assumed most characters will start with between 50 and 74 CP, children though being inferred to have between 0 and 24 CP). This can woefully weaken the character at the start of a game, though that makes sense since you are playing a child but also feels a bit out of place for a system that is meant to play a lot like your favorite anime.
GURPS
Memes aside, GURPS does indeed have rules for playing lolis, almost all of which are drawbacks. So while you can, you're not given any incentive to do so except the points any disadvantage gives.
Kids on Bikes
Kids on Bikes offers the players three age ranges to play as, one being child (the others being teen and adult). Each age range gives two stat modifiers and some sort of bonus. Children, in this case, heal faster and are generally better liked. These adjustments and such don't change the largely narrative nature of the game though and based on how stats are allocated you could be stronger, smarter or otherwise better than an adult in many cases.
OVA
Another anime-styled RPG, this one is even more open-ended than BESM but explicitly has rules for characters who are young or old in the "Ageism" weakness which has 3 ranks of severity. At rank 1 you are basically a young teen or someone past their prime, rank 2 you're a child (or simply look like one) or clearly and old ass, and rank 3 you're basically a kindergartner or someone so old they may well be on life support. These drawbacks tend to come up when you're dealing with people such as when you're trying to convince them of something like "there are monsters under the bridge." It's not exactly the best but the system is meant to be fairly open-ended so, like BESM, it works but could use some extra crunch to it.
Savage Worlds
In Savage Worlds, Young is a major (8-12) or minor (12-17) Hindrance, depending on just how young you go. In addition to the social penalty, you also lose attributes and skill points, which is bad because one of the big reasons to take an Hindrance is to gain more of those. It does however give a free Benny or two a session. The ability to trade a Benny for Power Point refill, and need for Edges (the other major reason to take a Hindrance) to know magic and improve it means loli characters can make very good spellcasters (or super science inventors, which is more or less the same as magic in Savage Worlds)
Star Trek Adventures
With a bit of a history of children getting into Starfleet (or at least getting caught up in the ongoing adventures of a ship or station as seen with Wesley Crusher, Jake Sisko and myriad of other kids/teens), it is no surprise that the core rulebook for this game had to make some passing mention of how old a character can be. The way they handled it? "As long as the character passes an exam to get into the Academy, they can be an officer." They do say the average age of a cadet is someone in their twenties, but they don't explicitly say you can't play younger either. So go out there and get a whole crew of rascals but beware any strange green energy beings telling you to murder all of the adults, okay?