Open-Ended Boss Battle

Let's be blunt: In most Boss Battles, there is only one "correct" outcome to the battle and the game will not proceed until it occurs. Get slaughtered by a boss? Unless that was the point, go back to your Save Point and try again.

But sometimes The Dev Team Thinks of Everything and allows the game to proceed regardless of whether the player won or lost the battle in question.

Obviously, failure to defeat this boss means the player loses out on the opportunity to collect Experience Points, money, or items associated with victory, but some games may secretly record this outcome for future reference, allowing it to have subtle effects later on.

The Monster Arena is a popular venue for hosting open-ended battles, although the outcome of these battles rarely extends outside the bounds of the venue itself.

Note that getting a Nonstandard Game Over or an ending from getting the unexpected outcome of the fight is not this trope.

Subtrope of Boss Battle. Contrast Hopeless Boss Fight, Heads I Win, Tails You Lose.

Action-Adventure

 * The final fight against the Oracle in Fahrenheit can be won or lost. In the former case, Lucas gets to fight the Cyborg later, while in the latter, you take control of Carla in an attempt to save Lucas--which you can also fail, giving you one of the three endings. The other two are received for either beating or losing to the Cyborg.

Eastern RP Gs

 * In Breath of Fire 3, although the player loses their first fight against Balio and Sunder, when Ryu battles them later to save princess Nina from abduction, the player may win or lose that battle without penalty.
 * In the first generation of Pokémon, the player's very first Mon battle was against their rival, and they could win or lose this fight without penalty or consequence, although the "Special Yellow edition" used this as a factor to determine which form The Rival would ultimately evolve his starting 'Mon into.
 * In the first Shadow Hearts game, Kouldeka, whether or not the player could defeat the game's Final Boss determined whether they earned the game's Good or Bad Ending, with the latter being considered canon. What makes this unique is losing is what gives the good ending.
 * In Chrono Trigger in the Kingdom of Zeal, the plot requires you to be captured. To this end, a Golem is sent to attack the party. While the standard fare would be to lose to the monster, you can actually defeat it (in fact it can be pathetically easy if you know how). Doing so results in the summoner using the "look behind you" trick then shooting you in the back. All winning does is net some decent EXP.
 * Except for the last Lufia game, the mad gods known as "Sinistrals" could only be killed with a magic sword called "the Dual Blade." However, defeating one in combat prior to obtaining it would result in a rare equipment drop, and a slightly altered cut-scene afterward where the Sinistral blinks, then uses non-combat cutscene powers to defeat your characters anyways.
 * Tales of Symphonia has a famous three-boss sequence at a spoilerrific important scene that goes regular boss -> Open Ended Boss Battle -> Hopeless Boss Fight. If you lose, the boss can't bring himself to finish you off . If you win, he can't finish you off because, well, he lost. Either way, this is where the Hopeless Boss Fight steps in to finish the job.
 * Golden Sun
 * In the first game, the Colosso competition can end in your Victory or Defeat of the tournament. Winning nets you an extra item and (if you export your save data) an extra scene in the following game. Loss still advances the plot, but with fewer goodies.
 * The battle against Agatio and Karst on top of Jupiter Lighthouse in the second game is also like this. After the battle is over, Alex shows up, heals the losing team, then the three of them run away, and the plot advances like normal. But if you win, you get a bunch of money and XP and a chunk of Dark Matter.
 * In the first Kingdom Hearts, when Sora first arrives in Traverse Town, Leon fights you in the town square. There's no penalty for losing, and it is very difficult to win (Unless you're aware of a certain flaw in your opponent). If you do win, Sora will delight in his win (and major EXP gain) and then promptly pass out. The next scene will have Leon praising Sora, instead of the usual "This is supposed to be the Keyblade master?!". You can also get an Elixir from Yuffie afterward.
 * Early on in Secret of Evermore, you're attacked by 4 Raptors. If you lose, your dog drags you to the nearby village and a NPC heals you. If you win, you get money and continue on into said village, where another NPC rewards you with an earlygame armor for free.
 * In Arc Rise Fantasia, there are summons that go to the strongest of two parties after certain battles. You can both win and lose these battles, and change the summons that you and your rival can use.

Fighting Game

 * During story mode, in Rival Schools, your team will encounter Raizo, as the third battle. You're meant to lose the match, afterwhich, a designated member of your team is taken hostage while the other two are brainwashed and made to do Raizo's bidding for the next two battles. Depending on the team, and their story, the effects of Raizo's mind control either wear off or you're restored to your senses after the second battle (following the Raizo encounter). If you manage to defeat Raizo, the game skips the mind control phase of the story and cuts directly to the battle with Justice High. You still lose a member of your team, however, as the story demands this happen. They're either kidnapped by Raizo, after the battle, or they proceed to Justice High on their own to speed up the team's investigation. By the time you arrive, you learn your missing teammate was caught snooping and is now a hostage.
 * Trapt: If you decide to fight the final boss, you get some interesting results, depending on the outcome. If you lose, Princess Allura/Alicea will get possessed by the demon without anyone the wiser. If you win, the Princess will vanquish the demon, and go back to the castle with her faithful knight.

Massively Multiplayer Online Games

 * In original Guild Wars, the bonus objective of the Dragon's Lair mission involves killing a major character. If you loose, your character is simply resurrected, and completes the mission anyway. If you win, the character's death has no effect on the story, and they still communicate with you at some later points.

Platformer

 * In the Mega Man Zero series, if the player loses a life out on a mission, they may be given the option to "give up", where the mission is a failure but the story continues without the mission's rewards.

Shooters

 * In the 1987 shooter Dragon Spirit, the game's Warmup Boss determined whether the player progressed in "Easy" or "Hard" mode, with "Easy" mode giving the player a stronger dragon, and "Hard" mode featuring two additional levels and the game's True Final Boss.
 * In an old arcade game called Ocean Hunter, you, at one point, encountered a shark, tougher than the rest, tearing at a scuba diver. If you managed to kill the shark before the diver died, you got an extra life -- but nothing happened if you didn't manage to save him.
 * In Action Doom 2 Urban Brawl after riding the subway you come up against two gang leaders; depending on whether you beat them or lose to them, you go down a different path and eventually get a different ending. Interestingly, it is losing to them that will lead to the best ending you can get at that point.

Western RP Gs

 * In Dragon Age: Origins, during a certain rescue mission the player is confronted by the powerful Ser Cauthien. If the player manages to defeat her, the mission is complete and they may return to their Quest Giver immediately. If the player is defeated, they are thrown into prison and must find a way to escape (and will fight her later).