Air's Kamio Misuzu. The girl knows that if she ever makes any friends they're both going to get sick and possibly die, and yet all she wants to do with her summer is play with a creepy stranger at the beach.
It's not that she knows such bad things will happen, it's more like she just can't seem to connect with others because her optimism and personality are just so...odd. And I wouldn't say Yukito is creepy, just a hot teenager new to the town that little Misuzu thinks might appreciate her.
To be more specific: Misuzu doesn't seem to have trouble getting to know the other girls or Yukito that whole summer, and she doesn't know about her curse, BUT she knows from past experience that as soon as as soon as she gets close to someone she'll have physical pain and mental panic attacks. A later flashback episode shows her ready to give up on even one happy summer before she sees Yukito on the beach and has a feeling he might play with her.
Nagisa from CLANNAD. She finds a very small and old apartment with almost no furniture "a very nice place".
Komari and Kud from Little Busters!. Komari's route was something of a Deconstruction, revealing the darker side to her idealism and involving a happy conclusion that she was now able to accept that sometimes bad things happen, but Kud was able to go through her entire route holding on to the belief that there are no bad people in the world, even while she was being chained up as a human sacrifice.
Fate/stay night: Emiya Shirou has had a rough life: He lost everything he held dear (including his memories) in a giant fire caused by the destruction of the Grail and was only barely saved by Emiya Kiritsugu who adopts him... and then dies 5 years later To the curse of the Holy Grail, having to live alone as a 13-year old with no friends or family, with only a guardian in the form of Fujimura Taiga, who doesn´t even live in his house. Despite that, he´s always friendly to anyone he meets.
Maya Fey rules this trope. Her parents left her when she was very young, she's framed for her sister's murder, her aunt tries to frame her for another murder, she's kidnapped and starved for ransom, Dahlia Hawthorne tries to kill her in another murder plot by her aunt, AND her mother is killed in front of her by her sister's (former?) lover. And yet she still manages to be the perky sidekick almost all the time.
Kay Faraday too, seeing as her childhood friends Edgeworth and Gumshoe at first don't remember her, her dad was murdered when she was young, leaving her an orphan, and Calisto Yew turned her father's proud secret legacy into nothing more than a murderer and thief. Sure, she gets upset, but she always bounces back.
As for Pearl Fey, let's see... Her dad left her when she was tiny, and in a span of less than ten years one of her cousins dies, the other is framed for murder three times and kidnapped once, her mother is sent to prison for framing said other cousin, she meets her favourite author who turns out to be her long-lost aunt, who then dies, she finds out she has two older sisters, one who's an evil, murdering spirit with no love for her and one who is suspected of killing her aunt and is later sent to prison for her part for covering up a murder. Despite all that, Pearl still remains an optimist with a belief in fairy tale endings, though she is quite emotional and open about her distress.
Regina Berry leads an incredibly sheltered life in a traveling circus, innocently sets up a prank that goes horribly wrong, causing her friend Bat to fall into a coma which he'll most likely never wake up from and rendering his brother Acro wheelchair-bound for life, which leads directly to the death of her father. And yet she still doesn't fully understand her situation until Moe takes her to watch Max's trial and forces her to realize the consequences of her actions.
Maggey Byrde has, in her own words, suffered every kind of accident, caught all sorts of illnesses, failed nearly every exam she's ever taken, and been framed for murder thrice, each time losing her job. Still cheery, though.
Trucy Wright has, by age 15, lost both parents, ends up supporting her deadbeat adoptive dad, rarely sees her much richer uncle, and finds out her biological dad died and was partially responsible for ruining Phoenix's career. Of course, she's a Genki Girl. Although Trucy might be closer to a far less drastic version of a Stepford Smiler since Phoenix reveals that her Pollyanna persona is, in some ways, just a front.
Phoenix: "Because I'm the only one who knows how she really feels... on the inside."
Athena Cykes continues the tradition, having repressed the awful memory of her mother's death and possible involvement, using the tragedy to motivate her and exonerate the person who took the fall in her place. She puts up a strong front of an outgoing girl with a lot of energy and enthusiasm to mask her internal problems. Even at her darkest hour, being led into custody, she insists on sending her congratulations to the person she was replacing (whom she had helped defend in court), even though he says her eyes were still puffy and red from crying on the bus ride over.
Battler of Umineko no Naku Koro ni is yet another male example. He steadfastly refuses to acknowledge the existence of witches (or at least refuses to admit that they're causing the murders) in a universe that is not shy in its attempts to break him. As a When They Cry protagonist, being a Determinator is a requirement. The universe REALLY pushes it, though.
Maria is arguably even more of an example. She's constantly abused by her mother who essentially wishes she was never born and thinks having a child stands in the way of her finding romance and living a happy life, and feels that her 'creepy' hobbies are an embarrassment and reflect on her poorly as a parent. Couple this with being constantly bullied by her classmates and looked down on by teachers and it's no wonder why she retreats into a magical world where she's surrounded by friends and happiness. Even when she watches her various relatives, in some cases including her mother, get brutally murdered right in front of her, she remains (albeit creepily) optimistic because she's friends with the Golden Witch supposedly committing these murders, and has been promised that even in death she'll be brought to the prosperous Golden Land, and even find a version of her mother who loves her unconditionally and always has time for her waiting there. Once you realize this and go back to the older story arcs, Maria's status as the resident CreepyChild lessens as it becomes apparent that this occult interest is the only thing keeping her from breaking down entirely from how horrible her life is.
Shiki Tohno from Tsukihimehad his family slaughtered (although he doesn't remember this) and survived a fatal accident. He then started seeing lines everywhere and if touches them, he will destroy whatever object that line is on. He was about to go insane until he met Aoko who helped him cope with his problem. Because of that, he appreciates life more. Enough that the one thing he will not tolerate are monsters that take lives from others.
Tanomura from Swan Song is another male example. He acknowledges when a situation goes south and reacts accordingly, but expect him to be cheerful in every other situation you encounter him in.