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Majora's Mask has a kidnapped princess that is the basis for reaching the first dungeon. However, you can finish the game without needing to free her from her prison.
Both The Legend of Zelda: The Wind Waker and its sequel, The Legend of Zelda: Phantom Hourglass, are kicked off with the kidnapping of a young lady — your sister in the first, and the actual princess in the second; much of the story centers around their rescue. In both games, though, the plot carries on well after you've saved the ladies in question.Four Swords also has very little plot beyond this.
The Legend of Zelda: Spirit Tracks both averts and lampshades this. While Zelda's body is stolen, she, in spirit form, sticks with Link. When she realizes there's a Big Bad to be defeated, she promptly tasks Link with the entire task, claiming that sitting around and waiting for the hero to rescue her is a "family tradition". Soon after, though, they discover that Zelda can be useful in combat and they work as partners from then on.
Shining Wisdom's first half is basically just to save the princess. The rest is stopping the destruction of the world.
The first two Ninja Gaiden games on the NES has Ryu defeating some form of Cosmic Horror to Save The World while also saving the CIA agent Irene Lew, who becomes his girlfriend at the end of the first game. In the third game, Irene is presumed dead in the beginning but is alive and doesn't require rescuing.
Fat Princess turns this into a Capture the Flag game, with the goal being to get to the opposing team's base and carry their princesses back to their side. You can hinder the progress by feeding the Princess cake, which will have her grow fatter and becoming heavier to carry.
"The Prince Gilgamesh wore golden armor and attacked monsters to save Ki in The Tower of Druaga." In this case though, Ki (pronounced "Kai") is not actually a "princess" per se, she's a shrine maiden in service to the goddess Ishtar (who lends her name to the game's sequel, "The Return of Ishtar").
The goal of Penguin Adventure is to find a golden apple to restore the health of the penguin princess. Because of Guide Dang It, it's easier to get the bad ending in which she dies.
In King's Quest III: To Heir Is Human, the Oracle of Llewdor first lets "Gwydion" know about about the three-headed dragon terrorizing the faraway land of Daventry, and the young princess sent as its latest sacrifice. The Oracle then drops the second bombshell (she's his sister, meaning he's the lost prince), and the third (his name's not Gwydion at all).
And subverted in-universe in King's Quest: The Floating Castle, when both the Big Bad and the princess he's holding captive think Alexander has this in mind, when he's really just there to save his father's soul. (Though he does rescue her anyway, both because she can help him out and because it'd be rude not to.)
Red Steel featured its One-Man Army hero tearing through the machinations of the Yakuza in an international battle to bring down their new, more violent and corrupt leadership, learning legendary sword techniques along the way, after they kidnapped his fiancé and killed her father, the previous and attempting-to-go-legal Yakuza leader at their engagement announcement dinner.
In World of Warcraft, a quest charges you with saving the dwarven princess Moira Bronzebeard from the emperor of the dark dwarves. It turns out she's pregnant with said emperor's child and doesn't want to be saved...
Also Daisy in Super Mario Land, one of the few games where Bowser isn't present. Apparently he only got a license to kidnap Peach.
Super Princess Peach inverts the trope - Bowser jacked the Vibe Scepter and sends his minions in to use it to paralyze THE ENTIRE CASTLE in emotional distress, using the chaos to imprison THE MARIO BROTHERS (arguably one of his smartest moves in some time)! Peach was Late to the Tragedy, and thus puts it on her shoulders to bail the brothers out.
Recently averted in Super Mario 3D World where Peach is now a playable character alongside Mario, Luigi and a Blue Toad. Though the game still sees you rescuing another princess, the Sprixie Princess.
In Sonic CD, Sonic's mission is to save Amy from Dr. Robotnik and Metal Sonic. Many feminist writers like Anita Sarkeesian have lumped Amy in with other well-known gaming damsels-in-distress, not realizing (or choosing to ignore) that she actually hasn't needed to be saved in any other games in the series, besides one part of Sonic Riders: Zero Gravity. In fact, she's much more often been a playable character, sometimes saving others.
Sonic Rush further averts this trope by introducing an actual princess character, Blaze, but having Sonic team up with her. In fact, Sonic and Blaze rescue Cream together in that same game.
To give an example of how this trope was so prevalent to the point of getting absurd, in the NES games Tiny Toon Adventures and Muppet Adventures, you are charged with saving rather strong and self-sufficient characters Babs Bunny and Miss Piggy, respectively. The latter example in particular is rather egregious, since it's far, far, far more likely that Miss Piggy would have to save Kermit from a bad spot (And she did, in The Muppet Movie. She'd also have saved him in the episode where the pigs took over the show, except she got offered a star position).
This is the initial premise of Eversion. The endings will make you wish you hadn't saved her.
Done in an antiheroic way in Wario Land: The Shake Dimension, where saving the queen is only Wario's secondary goal, with him being promised treasure at the end. In fact, at the end where Queen Merelda congratulates Wario, he stops her in the middle by grabbing her and tossing her out of his way.
At the end of Earthworm Jim 2, it appears that "having defeated the nefarious Psy-Crow, our hero, Earthworm Jim, has won back the heart of the lovely Princess What's Her Name." Except she's a cow wearing a costume. So is the villain. So is Jim.
The main plot of the first Chip 'n Dale Rescue Rangers game on the NES is to rescue Gadget. This notably doesn't start till after you've beaten the first level where the goal is to find a lost kitten (which it turns out was Fat Cat's way of distracting you), and the game goes on for three more levels after you rescue Gadget.
This is inverted in the PC game version, where Chip and Dale spend the game collecting screws so that Gadget can finish the Ranger Plane and rescue Monteray Jack from Professor Nimnul.
You have to rescue your love interest in Meat Boy.
The Clonk level "Dragon Rock" plays this unashamedly straight, right down to the evil mage-with-a-dragon doing the kidnapping. In "Tower of Despair" it's the king, and the dragon itself is the capturer, but otherwise it's pretty much the same.
Ghosts N Goblins and its sequels begin by showing Princess Prin-Prin getting kidnapped by some horrible demon.
The first Knightfall game has the protagonist literally drilling his way to Hell to save his princess girlfriend from the Devil.
The goal in Castlequest is to rescue Princess Margarita from Groken Castle.
Kickle Cubicle for the NES actually has four princess to rescue, one at the end of each of the lands.
A minigame/side-quest in Catherine involves playing a game-within-a-game at the Stray Sheep called Rapunzel. In it, you solve block puzzles much like the ones in the "Nightmare" segments of the main game, except there's no enemies, and instead of a time limit you have a limited number of moves with which to make a path to the top of a stack of blocks so your Prince Charming character can get to the titular Rapunzel.
Real Time Strategy
Lampshaded in Supreme Commander: Forged Alliance, in one mission you must fend off the alien attackers from Princess Rhianne Burke's personal palace, Mission Control tells you that "You have a Princess to save, Commander."
Role Playing Game
Dragon Quest I essentially has only two objectives: Save the Princess, and kill the Dragonlord. Unlike most Save the Princess games, however, you actually rescue the princess from a dragon (usually the first one you kill) in the Marsh Cave long before you beat the Big Bad.
Final Fantasy IX: Zig-Zagged at the start; your task is to kidnap Princess Garnet, but when you confront her, she actually requests the kidnapping - making it clear this is more of a rescue mission. Later dialogue with Regent Cid reveals that he had commissioned the kidnapping as a cover for the rescue, as taking Garnet away openly would cause quite a political stink.
Much later, Queen Brahne orders Garnet's execution, and Zidane and the rest of the party must storm the castle to find her before it's too late.
Final Fantasy X: At one point Yuna, daughter of High Summoner Braska - a princess in all but title - is kidnapped by the Corrupt Church and kept in the castle-like (and gigantic) Bevelle Temple. Oh, and she's forced to marry Seymour. The party arrives to intervene. Subverted when Yuna reveals she was using the wedding to try and Send Seymour, already had an escape planned, and the party's intervention has royally screwed up her plan by giving the villains hostages.
Final Fantasy XII: To recruit Ashe - your fifth permanent party member, and you guessed it; a princess - you have to cross an enemy's heavily-guarded airship to reach her cell after she'd been taken into custody a good 3-4 game-play hours ago.
There is a mission in Valkyria Chronicles where your squad has to quite literally save the princess after a kidnap attempt.
Your first real goal in Knights of the Old Republic is to rescue Bastila from the swoop gang that recovered her from a crashed escape pod and is now offering her as a prize in an upcoming swoop race. Then, two-thirds of the way through the game, Malak captures her... (Of course, the movie whose universe KotOR is set in had Save the Princess as a good chunk of its plot as well.)
Oh, so very, very much subverted in Live A Live: Orsted sets out to rescue his princess bride-to-be in an opening obviously inspired byGhosts N Goblins. By the end of the chapter, she thinks that his friend was the only one who truly deserved her, even though he's turned evil, and commits suicide so that the two of them can be together forever. This leads to Orsted becoming a demon of pure hatred and the Final Boss.
Your first real objective in Chrono Trigger is to retrieve the girl who has fallen through a time warp; and sure enough she turns out to be a princess. Subverted in that when you get back, you're put on trial for kidnapping her in the first place. You end up having to dive through another time warp in order to evade the guards, and that's where the real adventure begins.
In Faria, the first mission you receive is to rescue a princess from a tower. However, you can't marry her because you're a girl. Moreover, this princess is a fake, and you find the real princess in a later tower.
In A Witchs Tale, Liddell must save all six princesses before she can meet Queen Alice.
In Resident Evil 4, Leon pretty much spends the entire game going through the Big Bad's trap-filled castle, fighting off undead minions, so he can rescue the President's daughter, Ashley. At the end of the game, Ashley offers him a lot more than just a Smooch of Victory, but given he's part of the secret service in that game, that would obviously not have been a good idea, and he rightly turns her down.
Almaz: I must save the princess! Sapphire: *Groin Attack* Almaz: The princess! She touched me! I can die happy now...
Disgaea 3 also looks at it more seriously with the reasoning behind its Inversion: Princess Sapphire has seen far too many people go off and die all in the name of protecting her. So she became a Badass Princess capable of destroying anything that might kill a hero.
The Cursed Memories take involves Axel kidnapping Taro and Hanako as bait for the 'wild tribesman' Adell so he can "rescue" Rozalin.
Fire Emblem: The Sword of Seal gives you two "save the princess" missions, one with Princess Guinevere of Bern (the half-sister of the Big Bad, who becomes your protegé) and the other with Princess Lilina of Ostia (who joins your troupe as a Magic Knight as soon as she's rescued)
In Genealogy of the Holy War, Starting in Chapter 10 you have to find Princess Yuria... in classic style she's in another castle every time you conquer the previous castle... this lasts until the final part of the final chapter, where she's That One Boss hanging out in your way.(Note that she's That One Boss who you can't kill (or the game will become practically Unwinnable- she's needed for the Final Boss fight.)
In The Sacred Stones Princess Eirika is the main character of the first several missions, and is at least initially most concerned with saving herself. Depending on what route is taken there may also be a mission where Princess Tana must be rescued.
In addition, after getting to safety (And Saving Tana) Eirika sets out on a quest to save her brother making this 'Save the Prince'. Except that, in the end, he's the one who rides to her rescue.
In Shining Force 2 the Shining Force must rescue Princess Elis from Dark Sol. But in The Sword of Hayja, the trope is inverted and the object is to rescue Prince Nick.
Examples from other media:
In a Save The World climax of Mahou Sensei Negima! main objective was to free Asuna (revealed at this point as a princess of a fallen magical kingdom). Anya even lampshades that for Negi rescuing her is most important, and saving the world is just a bonus. Through zigzaged in that there seems to be more familial than romantic feelings between them (and he is even related to her)
Subverted in Magic Knight Rayearth. Turns out the real mission is to kill the princess at her own bequest.
While technically not a princess (unless you count her as one for being the daughter of the Fey clan's leaderess), poor Maya Fey in the Phoenix Wright: Ace Attorney series has to be saved from death four times! The first two involve Maya being accused of murder and you have to get a not guilty verdict to save her. The third occasion has Maya kidnapped by an individual who uses this as leverage to force Phoenix into taking a case. The last case has Maya being trapped on a freezing mountain. And things just get worse from there...
The game that Radd is from in the webcomic Kid Radd has this plotline. Interestingly, due to the premise of game sprites as sentient beings who are created for the express purpose of being in games, Radd has no idea he even has a girlfriend until the narrator tells him so.
Swap out "Princess" with "CEO and former Magical Girl Warrior", and we've got the main plot for Last Res0rt. Of course, both halves of the rescuer/rescuee equation are female (and said Rescuee is reasonably capable of saving herself), so...