Plenty of anime feature characters who either think they have amazing talents and supernatural powers or believe they're being targeted by an evil organization or government conspiracy. These individuals are referred to as chunibyo anime characters.
Chunibyo roughly translates to "eighth grade syndrome," implying the behavior is analogous to middle school—but delusional anime characters can exist in high school and beyond.
A chunibyo character might believe they're an alien, or that they have a powerful eye they must disguise with an eyepatch. However, they can also entertain mundane delusions, like believing they're proficient at writing novels or racing bikes.
No matter how their chunibyo characteristics manifest, these characters can add an interesting twist to the stories they exist in.
Rikka Takanashi is the poster child for the concept of chunibyo—in fact, she's the protagonist of an anime called Love, Chunibyo, & Other Delusions.
According to Rikka, she's a sorceress who thinks she's possessed by something called the "Tyrant's Eye." Anyone who sees this mystical eye – which can reveal a person's true destiny – unwittingly enters into a contract with Rikka. She also uses her umbrella to cast a variety of "attacks," and believes death is reversible.
That last point is the reason she clings so tightly to her beliefs. Her father died, and she can't accept it, so she uses her fantasy world to protect herself from pain, and to convince herself she can bring him back.
Megumin actually does live in a fantasy world and possesses mystic abilities, most notably her explosion magic. However, she still exhibits some exterior chunibyo traits, making her a cute example of the trope instead of a sad one.
Megumin wears an eyepatch she claims seals away her amazing powers, but she's also quick to admit it's a joke if anyone seems to actually believe her. In reality, she just wears the patch because she thinks it looks cool.
Rintaro Okabe is an interesting example of the chunibyo trope because he's not necessarily delusional. He pretends to be working against a secret, nefarious organization called SERN, and often claims to have special powers or knowledge he doesn't actually possess.
While he seems like a person with severe chunibyo, he's actually acting this way to amuse his childhood friend Mayuri, who is deeply depressed following her grandmother's death.
For the most part, Kobato Hasegawa is an ordinary middle school girl, but she has a couple of seriously strange traits. One of those is her belief that she's actually an ancient gothic vampire named Reisys VI Felicity Sumeragi.
She got this idea after watching her favorite anime, Kurogane no Necromancer. While she tries to speak in a vampiric accent, she often forgets to do this and reverts back to her Kyushu dialect, much to the amusement of her brother Kodaka.
Her fixed beliefs cause her to have trouble making and keeping friends, a struggle shared by many other characters in Haganai.