Parodied in the Animaniacs movie, where a sickly Dot needs money for an operation. At the end of the movie, it turns out that all the operation did was give Dot a beauty mark to make her even cuter.
Subverted on The Batman. A scientist working for Wayne Industries claims to be studying bats so he can cure his niece's deafness. Bruce Wayne goes to see the girl and give her a hearing aid... and she turns out not to be deaf. The scientist was actually studying bats because he was obsessed with Batman, and he eventually turned himself into "Manbat".
This is the reason why Victor Fries, better known as Mr. Freeze, constantly has to commit crimes in most of his appearances in Batman: The Animated Series and related media: His wife, Nora Fries, is suffering from a terminal disease, causing him to put her in cryogenic stasis until his research allows him to develop a cure for her ailment. Unfortunately, in order to do so, he ended up having to illegally appropriate research materials from Ferris Boyle, the Corrupt Corporate Executive of GothCorp, resulting in Victor being kicked into his own formula and deformed into Mr. Freeze. In his next appearance, he was taken by a deranged theme park founder and has to help this person in exchange for allowing him to get Nora's miraculously survived cryonic tank. He stopped only when Batman pointed out that this is not what Nora would have wanted.
Played straight in Jem: In the three-part "Starbright" storyline, the Holograms keep on going with a movie shoot into which the Misfits have bought themselves because they need money for an operation to save Starlight Girl Ba Nee's sight. Ultimately the production splits into rival films, and when the Holograms' film is a hit, the money is raised and Ba Nee is saved.
In the Christmas SpecialSanta Claus Is Comin' to Town, Kriss Kringle gets the bright idea to enter locked houses through their chimneys largely because he has to deliver a toy Noah's Ark to "Susan, a tiny little girl who was very, very sick".
Little Marcie from Adventure Time in the episode "Simon and Marcy." The episode's plot has her friend Simon trying to get her chicken soup in the post-apocalyptic wreckage. The original plan for the episode was for her to be deathly ill and Simon to be searching for medicine, but they had to tone it down to get by the radar.
Claire from Stressed Eric is allergic to everything, even to things that are supposed to help her allergies. Taken Up to Eleven with her nerdy Truman-esque friend Julie, who is allergic to more things! Even air will stop her!
Ruby Gloom has Malady, who is said to suffer from every physical ailment you can think of.
Played With often with Kyle. In the 15 seasons he has needed a kidney transplant, said to suffer from type 1 diabetes, nearly died from an infected hemmroid and just generally is shown getting sick with a far greater frequency than the other boys, Kenny included.
Strangely enough, usually averted with Kenny as he almost always died as a result of violent mishaps. The only times he succumbs to a disease is from an STD he picks up from the elementary school slut.
Rudy Steiblitz from Bob's Burgers has severe asthma but remains very active even when it's dangerous for him to do so. He's a bit of an Thrill Seeker.
Invoked and subverted in Sealab 2021: Sealab is visited by an ill boy named Griff in conjunction with the Final Request Foundation. Griff's actually a con artist bilking the foundation by pretending to be sick in order to get access to their bank accounts. He ends up eaten by a shark.