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There's quite an argument to be made for this trope being Older Than They Think; many of these character traits are actually found in 19th-Century Western literature (which was consumed voraciously once Japan opened trade with the world). The comparison isn't always exact (generally speaking, Victorian-era authors had rather different ideasabout how to use this character type), but this character type actually shows up quite a bit more often than you might think.
has Cosette. The Animated Adaptations that have aired in Japan over the years seem like they might've cranked the "moe" meter for her up to 11... but, uh, nope, that's pretty much exactly how she was described and acted in the book. Especially once she starts getting abused by the Thenadiers, even the most stoic among us pretty much want to scoop her up and give her a big hug and tell her it'll all be okay. Valjean then does this, and you can't help but cheer. This fades a little when she gets older, but Victor Hugo still makes her seem quite vulnerable.
Cosette's mother, Fantine. She gets knocked up and abandoned, debases herself in every way to help her daughter, and ends up dying of several diseasesat once. Or how about Eponine in the second half, who ends up destitute, uneducated, and in love with Marius but with no way to really express it, complete with Verbal Tics? (Not to mention shot dead pointlessly?) Really, the only major female character who doesn't have some of these elements is Madame Thenadier.
Also, you'll note something about the above: "Anime Adaptations". There have been four different adaptations of Les Mis produced in Japan, mostly from 1978-1979, and every single one featured a significant focus on Cosette (usually as a little girl, but sometimes going into the latter parts of the novel). The 1970s ones were widely watched, and a savvy reader can imagine how Cosette was portrayed in these shows (which is to say: close to her novel depiction). Even though it's easy to identify when the trope as shown here "coalesced", there is a very, very good argument to be made that the current Japanese moe wave and its idolization all traces back to Cosette Fauchelevent, in a kind of century-spanning Germans Love David Hasselhoff situation.
Victor Hugo really had a thing for this kind of character. Esmeralda also fits this type, being an ingenue and woobie who goes through a horrendous Break the Cutie process until she dies.
Charles Dickens also dipped into this territory on occasion; Tiny Tim is another excellent example of this trope (and, in a rare case, applied to a male - even in the 19th century this was usually done to girls). In fact, part of the narrative purpose of Tiny Tim was to be moe in-universe to help Scrooge feel bad about the fact that he'd inevitably die unless Scrooge changed his ways.
Of course, Charles Dickens being, well, Charles Dickens, he then immediately set about subverting the trope. The Old Curiosity Shop featured the character of Little Nell Trent, who was moe turned Up to Eleven. The craze surrounding this girl is legendary; Dickens essentially made entire countries feel protective about Nell. In the end, she dies in the most ridiculous, overwrought manner imaginable, and Dickens did this deliberately in order to parody the overly-objectified young females of the day in fiction. Yes, that's right, Dickens essentially deconstructed moe a hundred and fifty years prior to Cardcaptor Sakura airing. How old do you think the trope is now?
Oscar Wilde: One must have a heart of stone to read the death of little Nell without laughing.
An even better deconstruction that he did was Dora from David Copperfield, a vulnerable Adult ChildManic Pixie Dream Girl barely able to take care of herself. David falls in love with her and marries her, but finds that she's completely useless at helping him around the house. It gets to the point where he's more of her caretaker than husband, and though he still loves her dearly, it takes an enormous mental toll on him.
Tess, the eponymous protagonist of Tess Of The D Urbervilles is another example. Put through so much pain that you just want to give her a hug, not to mention being chaste and pure (at first, anyway). Like a lot of examples from this time period, though, the end result ends up a bit tragic.
O. Henry loved characters like this. Half his female characters match the trope almost exactly.
Paul Féval has some characters who might qualify, or at least have some shades of it.
In John Devil, Gregoryi Temple's Daughter might qualify.
The Black Coats has Blondette (not her real name) in Salem Street, Valentine has some shades of it in The Invisible Weapon but she's also pretty independent. And maybe Irene Carpenter in The Companions of The Treasure. But the meta example of that series would be the Colonel's granddaughter Fanchette.
The first Dresden Files comic, Welcome to the Jungle features Willamena "Will" Rodgers. Meganekko glasses: check. Twin braids: check. Deer in the headlights look: check. Wants to be under the hero's protection: check.
Speaking of Dresden Files, there's the Archive; perfect memory of everything written since before ever, anywhere, and holds all the accumulated memories of the previous Archives, as it's passed down from mother to daughter when the mother dies, and Ivy, as the current Archive was named by Dresden, had the unfortunate luck of having it all pushed unto her when she was a child, and her powers rival those of the Fae Queens. And when Harry met her the first time, she was all business, until she met Mister, Dresden's cat, and she went into a full 7-year old mode. And in Small Favor, when she saw otters, she went into full 12-year old mode.
Harry's daughter, Maggie, is another example. The poor kid has been through a lot, what with being kidnapped and almost used for a ritual that would've annihilated her entire bloodline. Her dad also avoids her out of guilt. When Maggie finally does talk to Harry, her interactions with her dad are adorable and bring some much-needed heartwarming for the fans.
In the Harry Potter books and movies, Luna "Loony" Lovegood is considered by some to be moe, even causing Anonymous to have the unusual reaction of "I want to date her and meet her parents" rather than "I want to have violent, squicky sex with her, then kill her"
Luna is certainly Adorkable and innocent, but there's much more to her than she lets on, and she does have an acute and realistic understanding of the world behind all that dreaminess, perhaps more acute than most people's. But she does get this treatment a lot from the fanbase, which unanimously adores her along with Neville.
For that matter, Bella herself. No self-esteem, clumsy, likes being stalked, has no other purpose in life than to serve her husband and kid? American Moe to the max.
Some commentators have described Twilight as the Western, heterosexual equivalent of yaoi manga in its relationship dynamics, and it's fairly common to give the Uke moe qualities.
Bonnie McCullough from The Vampire Diaries book series is the absolute personification of this trope.
Bertie Wooster frequently gets this treatment in fanfiction. He is entertainingly ditzy, goodhearted, and frequently in (temporary) distress due to the wacky Comedic Sociopathy-fueled schemes of his friends, and many female fans find him rather huggable.
Note that he has this quality in-universe, which is why he's such a Chick Magnet.
"I once consulted a knowledgeable pal," I said, "and his theory was that the sight of me hanging about like a loony sheep awoke the maternal instinct in Woman. There may be something in this."
Walter Plinge from Maskerade as well. He's just so earnest in his gawky misfit-ness. Totally subverted by the end. You can call Ghost!Walter a lot of things, but Moe isn't one of them.
Veesey-koosey from Corwainer Smith's short story "Think blue, count two" is a passenger in suspended animation aboard an interstellar vessel who as been selected to be part of the emergency crew not because she was good at anything but because she has a daughter potential of 999,999, meaning that anyone older than her would accept her as a daughter after a few minutes of relationship and because of that that person would be extremely motivated to save her life and hence the ship.
George R. R. Martin's A Song of Ice and Fire specifically invokes this with Shae during Tyrion's trial for murdering Joffrey; when she tells a (totally untrue) story of how he horribly abused her, the text says, "As the tears rolled down that pretty face, no doubt every man in the hall wanted to take Shae in his arms and comfort her," which is practically the definition of Moe. The circumstances make it one of the most brutal subversions of this trope you're ever likely to encounter.
To the people that like her, Sansa Stark. She is so sweet and in such a bad situation you just can't help but want to take her away from King's Landing and bring her back home. Her sister Arya might count if she weren't so disturbing.
Tommen is the most adorable little boy in this side of a Narrow Sea. Too bad a maegi prophesied his death.
Prim from The Hunger Games. Her sweetness and innocence is a little exaggerated by the main character, who is her older sister and would do anything to protect her. She becomes less of a Moe during the events of the second book, when she takes up to helping their mother treat starving and injured patients, and thus has to grow up too quickly.
Invoked in The Hostby the choice of Wanda's final host, who is so petite and pretty that other people will naturally want to protect her and not suspect her of anything.