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Brandon Higsby on As Told by Ginger. This often makes him the butt of Carl and Hoodsey's jokes.
Aang early on in Avatar: The Last Airbender. In the first episode, when presented with the knowledge that he has been frozen for a hundred years, a world war had broken out during his absence, and everyone he ever knew or loved is dead, he comes to the conclusion that meeting his crush more than makes up for it. His blissfully carefree nature gradually wears down over time, culimating in mid-second season where Appa is kidnapped, he has a Freak Out and spends the next couple of episodes trying to find ways to cope. One way is him hostile and overly violent, the next episode has him trying to be completely emotionless.
Uncle Iroh also counts. His son was killed in a battle he lost, his reputation was destroyed, he lost his rightful place on the throne and was banished from his home, his beloved nephew treats him like dirt, and yet he not only remains patient and cheerful, but is eventually able to help his nephew solve many of his personal problems without even breaking a sweat.
The titular character from Disney's Cinderella. She puts up with the demanding work her stepmother and stepsisters bullies her to do constantly, all with a cheerful smile and a pleasant attitude. It was only when they tore her dress and destroyed any hopes of her going to the ball that Cinderella broke down.
The titular character of Eek! The Cat never loses his optimistic outlook or his Catch Phrase "It never hurts to help". Even though every time he utters it, it inevitably hurts. A lot.
Lila from Hey Arnold!. Don't agree? Go watch the episode where we learn about her home life... just... watch...
Butters from South Park. This is actually mildly subverted in his "Very Own" Episode, which depicts his parent's anniversary and their annual trip to the restaurant Bennigan's. His mother asks him to tail the father to see what he is getting her, Butters finds out his dad is gay without realizing it himself (he describes what he saw and his mother puts it together), driving his mother mad, and prompting her to kill him and commit suicide. Both fail, but she doesn't know that the former did. The rest of the episode is the parents pretending that a stranger kidnapped their child, and when Butters shows up and everything is admitted, he is fine, saying "When I have a chipotle blu cheese bacon burger at Bennigan's, I'll forget all about my dad being queer and my mom trying to kill me." "Really?", he is asked by the other kids. "No, I'm lying." (His happy face barely cracks, though.)
The Oblongs — Living in denial, played for all the black humor the writers can get out of it. Bob Obling, the limb-less husband and father is the straightest examples of the trope.
The character, Daria, is like this in The Princess and the Pea. She remains cheerful in spite of doing all the chores on the farm as well as taking care of her lazy, unloving step parents. This is very evident when she prays for what she's thankful for before going to sleep in the chicken coop she sleeps in every night.
Ned Flanders from The Simpsons. Though he did snap in one episode.
In the Animated Adaptation of Where's Waldo?, Waldo would gleefully and knowingly walk into perilous adventure with the cheerful demeanor he always wears in the books.
Almost to the point of Obfuscating Stupidity at times. There was one instance where he was confronted with a vampire, and, without changing his expression or even blinking in surprise, simply withdrew from his pack a jar of his family's special pasta with extra garlic, gulped it down in one gulp, diliberately gave the vampire a huge dose of his now horrendously garlicky breath (Hhhhhhhhhey, there. Hhhhhhhhhhave a nice daaaaaaaaay?), then just happily went on his merry way as the vampire fell over, defeated.
Like the above comic example Miss Martian as seen in Young Justice, only here she is supposedly Martian Manhunter's niece.
On Goof Troop Goofy ends up physically injured, screwed over by his neighbor, or unappreciated by his son on a regular basis, and has a disposition so sunny it can really only be compared with that of the remarkably fortunate Pistol. However, he does occasionally have breaking points, especially in the movies.
Greg from Over the Garden Wall. No matter what dangers they face, he's always cheerful (or, occasionally, snarky). Even when Wirt convinces him that their troubles are all his fault, Greg doesn't dwell on it—he simply gets to work trying to fix everything. The Beast even seems to realize that he can't push Greg toward the Despair Event Horizon like he can with Wirt, and tries to freeze him to death instead.
Ms. Frizzle, the teacher from The Magic School Bus. Even when the bus and her class are getting baked into bread, assaulted by predatory animals, or attacked by a robot while on their crazy field trips, she more often than not has a smile and cheerful demeanor as she deploys a quick, magical way to get them out of the situation.