Eyelid-Pull Taunt

This is the act of pulling down one lower eyelid and sticking out the tongue and saying "Beh-da!" (Japanese) or "Nyaaaah!" (English dub). A highly ritualized expression of disdain or disapproval, used exclusively by children and immature adults. Basically the same as the Western act of sticking out the tongue, cranked up a few notches, thought not as offensive as Flipping the Bird.

In Japan this is referred to as akanbe (a corruption of akai me, "red eye"), and it's called mon oeil ("my eye") in France and Quebec. This is in fact a Franco-Japanese cultural trope, but you'll see it everywhere in anime. Easily mistaken for the Eye Am Watching You gesture, but it's entirely unrelated. See also Blowing a Raspberry.

Anime & Manga

 * Princess Mashiro does this to a trio of castle guards in the first episode of Mai-Otome, as seen in the image above.
 * Ranma and Akane give each other the Red Eye fairly frequently in Ranma One Half. Ranma gave once the Red Eye to Principal Kunō with his big toe, as his arms were tied up.
 * Usagi and Rei's relationship in Sailor Moon is largely giving each other red-eye.
 * Ryunosuke and Yoshimi give the Red Eye to Arisa and Kyoko at the end of the Beach Episode of All Purpose Cultural Cat Girl Nuku Nuku.
 * Happens at the end of the Haruhi Suzumiya episode "Someday in the Rain" by Haruhi. She also does it on the cover of the sixth novel, and the back of the DVDs in a chibi form.
 * In Bubblegum Crisis, Nene Romanova uses her powered-armor's laser to scrawl a picture of someone giving a red-eye on a sidewalk, as a gesture to the military who are tracking her by spy satellite, just before ducking under a bridge to drop off her armor in a truck and walk off as a perfectly normal girl. (This must have shamed said military a great deal, because we never see them again attempt to discover the Knight Sabers' identities this way, despite them not being very subtle about heading to and from their base.)
 * A couple times in Ouran High School Host Club Hikaru and Kaoru do this at the same time (in opposite eyes, no less).
 * Spotted in Shaman King.
 * Ino likes to do this in Naruto. Orochimaru does it once or twice, but with him it's just creepy.
 * In Full Metal Panic!, Kaname does this to Sōsuke when she ditches him (temporarily) at the train station.
 * Happens in Pokémon occasionally.
 * An Aerodactyl does this to Ash's Charmeleon just before he evolves into Charizard.
 * A Phione does this to Jessie in season 12.
 * Team Rocket did this on multiple occasions in season 1.
 * does this to Reinforce Zwei in the ending sequence of Magical Girl Lyrical Nanoha Striker S.
 * Mylene gives Basara the Red Eye as she drives away with Basara's Transforming Mecha to have it repaired in an episode of Macross 7.
 * Dita gives Jura the Red Eye in an episode of Vandread.
 * Viola does this in episode 7 of Kiddy Grade.
 * Happens in Dirty Pair occasionally (both in the anime and in the comics by Adam Warren).
 * Goku does this a lot as a child in Dragon Ball, as does present Trunks later on.
 * Hanaukyo Maid Tai La Vérité episode 7. Yashima Sanae has been insulted by Ryuuka, who insinuates that she's giving up on the expedition. As Mariel and Yashima Sanae walk off hand in hand, Yashima gives this to Ryuuka.
 * Squid Girl does this when Goro tells her and Takeru they're not supposed to be sitting in his lifeguard chair.
 * Yu-Gi-Oh: Jounouchi does this to Yami Malik after he reduces the latter's LP to 0 in the Battle City semifinal match up determining duel.

Films -- Live-Action

 * Apone does this to Hudson in Alien S "Look into my eyes"
 * Achilles does this to Alexander in the live-action Mecha movie Robot Jox

Music

 * Shown on the album cover for Yuko Miyamura's first album, Kenka Banchō.

Video Games

 * Mega Man's taunt in any crossover game he's featured in.
 * There's an emote for this for The Voiceless cast of the second Snowboard Kids game.
 * The pirate Briggs in Golden Sun: The Lost Age, while making off with a nice ship, is chided by his wife into doing what he apparently talked about all the time while in jail -- give your party the Red Eye. He was apparently on the verge of chickening out. He's stolen the ship and is well on his way out to sea, and he's almost too scared of Felix for a parting gesture.

Web Comics

 * Ariel does it in Drowtales as she leaves her "room" (really a cell that was formerly used to hold disobedient slaves). The author is French-Canadian.

Western Animation

 * Tiny Toon Adventures: Appeared in the They Might Be Giants "Istanbul (not Constantinople)" video of the music video episode. Considering how the episode was sourced to Tokyo Movie Shinsha, it may have been thrown in by the animators and kept.
 * In Teen Titans, when Control Freak has been bested by the Titans East, he cries foul, zaps himself into the TV and departs with this gesture, leaving the Titans East to try and figure out what the hell just happened.