Super-Persistent Missile

The munitions equivalent of the Super-Persistent Predator, this heat-seeking or similarly guided missile just will not give up. High-Speed Missile Dodge? It'll just come back for another try. Hide behind something? It'll go around. Try to mess with its targeting? It's not fooled. You can't even outrun it, because it has truly boggling range. Basically, when a Super Persistent Missile is chasing you, the only things you can do are shoot it down or die.

Homing Boulders are supposedly unguided projectiles with this capability.

Anime & Manga

 * In Fairy Tail, the missiles that the Doroma-anima fires at Natsu.

Film

 * The seeker torpedo that Jango Fett fires at Obi-Wan in Star Wars: Attack of the Clones.
 * A really persistent one in Hot Shots, later used by Topper himself to destroy Saddam Husseins Lair/Nuclear Plant.

Live Action TV

 * The bazookoid missiles From the Red Dwarf episode Polymorph, which were actually locked in a room and kept flying around in there for a week, until the heroes opened the door just in time for them to hit the Polymorph and kill it.
 * An episode of Star Trek: Voyager featured a star ship sized missile intended to wipe out a planet. The ship/missile had on board weapons to wipe out any ship that tried to intercept it, and an AI system that compensated for any interference in its mission, including orders by its programer to deactivate.

Newspaper Comics

 * Happens a lot in FoxTrot with Jason and Marcus' toy rockets. Even though the rockets are completely unguided, and in fact are supposed to go straight up.

Video Games

 * The higher-end missiles from Elite II: Frontier and Frontier First Encounters cannot be stopped by any kind of ECM. They still have only 60 seconds of fuel, but if you are in range, they WILL hit you.
 * In Freelancer, Cruise Disruptor missiles have incredible range, blinding speed, and aren't fooled by chaff.
 * The QAAM from the later Ace Combat games is one of the only missiles in the game that can turn around, and is incredibly hard for the AI to dodge.
 * They were literally impossible to dodge in Ace Combat 04, making all boss fights laughably easy after you unlocked a plane equipped with them. Thank the skies the enemies never used them due to faulty AI.
 * Ironically, the one thing (aside from you) that can dodge it in that installment is itself a missile. One that's persistent for a different reason entirely.
 * Cutscene missiles do this too - in a zigzagged example, Bartlett gets shot down by a SAM even after performing tricks that would shed most other missiles in the series, after drawing it off of Edge.
 * In Escape Velocity, it is all but impossible to avoid missiles without a Missile Jammer or a sufficiently thick asteroid field.
 * The game N+ features banks of missile launchers that fire nasty homing missiles that never stop trying to kill you, ever. In the event you get one to smash into a wall, another one will just be launched.
 * SAMs in Command & Conquer Tiberian Sun. They will chase your helicopters across the entire map and they will hit since you don't have time to land before they catch up with you.
 * Occasionally occurs in World of Warcraft, but only visually. If you are on a high-speed flying mount, missile effects (magical and physically alike) are just about as fast as you, or even slightly slower. The damage itself always lands, but the graphic effect may end up chasing you for ages until you let it catch up to you.
 * Not a missile per se, but you can buy Toy Zeppelins from the few toy vendors ingame. These are used for casual fun, and once thrown to other players, it leaves your inventory and appears in theirs (assuming they have enough space). The Toy Zeppelins are very possibly the most super-persistent "missiles" in gaming history, as they will follow you very slowly wherever you go - there have been cases where a Zeppelin has been thrown to a player, the player then teleports to another continent, and finds a Zeppelin approaching him four hours later.
 * Mario Kart has two variations on this... the Red Shell, which homes in directly on the next person ahead (and in later games, will maneuver itself through the course until it finds its target). It gets turned Up to Eleven with the Blue/Spiny Shell, which fires relentlessly toward the leader with practically no way to stop it (and in later games causes a huge, blue explosion to damage nearby drivers as well)

Western Animation

 * Happens to Wile E Coyote and The Road Runner, a lot.
 * In the Dastardly and Muttley in Their Flying Machines episode "Follow That Feather", the Vulture Squadron used a feather-seeking missile to pursue the pigeon.
 * In Superman: The Animated Series, Mr. Mxyzptlk turns himself into a Super Persistent Missile (with a kryptonite warhead) to chase down Superman. Superman leads him on a merry chase, causing Mxy's smoke contrail to write his own name in the sky, backwards, twice (which Mr. Mxyzptlk agreed to leave forever if done).