Typically, games let you make choices. Some games let you make choices that have significant effects on the game world. The Big First Choice is an early player choice — sometimes occurring before proper gameplay even starts — that has a massive effect on the game.
You escape the enemy's dreadnought aboard your commandeered shuttle, with the rescued rebel leader in tow and mere seconds to spare before the explosive charges you planted crack the monstrous ship in half. Not a bad job for the tutorial level. As the adrenaline wears off, the rebel leader congratulates you for your courage and offers you a chance to work alongside her to help free the galaxy from the Dark Order's clutches. But your hero would have to give up his grand life as a space mercenary, an outlaw-for-hire: certainly he's no friend of the Order, and he's more than happy to fight them on his own time, but he chafes at taking orders from anyone.You ponder the decision for a few seconds, then have your hero turn down the offer with characteristic nonchalance. You then spend the next twenty hours of this epic space opera being inadvertently dragged into one conflict after another, meeting the rebels under awkward circumstances, and generally making life hell for the Order. Finally, when all's said and done, you get to put a bullet between the eyes of the Shadow Prince, collect your reward and fly off into the sunset.Feeling good about your victory, you go online to discuss the game. But what's this? You don't remember an infiltration mission aboard a satellite. And a romantic subplot with the rebel leader? How come no one's talking about the MX-6000 railgun that got you through the final stages, but they're all gushing over that multi-missile launcher that the Elite Mooks had towards the end? Certainlyyoudon't remember getting one.After skimming a few threads, you can't help but wonder: did you even play the same game as everyone else?
The answer: not quite. As it turned out, a lot more was riding on the rebel leader's innocuous first question than you ever could have guessed. By making one choice or the other, you determined the entire course of the rest of the game.
Not every game is quite as extreme as this example, but the Big First Choice (or second, or third) is a common way of extending the life of a game by making the player's choices at key points have a dramatic effect on the way the game plays out - perhaps even the way the game plays, period. Multiple playthroughs are absolutely necessary to wring 100% Completion out of games that feature such choices: sometimes the different paths will converge again at the end, but it's just as likely that each individual choice will lead to a different ending.
One method of Story Branching. Compare Multiple Game Openings, where the the story branches even before you make the first in-game choice, and Last-Second Ending Choice, where you play through most of the game before a major and irrevocable plot split.
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The first thing you do in Nox is pick your Character Class, which also decides which one of three largely different storylines the game will follow.
Played with and lampshaded in Super Paper Mario. Your first choice of the game is whether you will accept the challenge of saving the world. If you say no three times, you don't save the world and automatically get a Game Over.
At about the end of the first third of Indiana Jones and the Fate of Atlantis, the game splits into one of three possible paths depending the player's choice: the combat-centric "action" path, the self-explanatory "puzzle" path, or the "team" path where Indy travels together with Sophia. These paths all converge at the last act in Atlantis, and beating all of them is required for the maximum Indy Quotient points.
In the Adventure scenario of Clonk's "Metal and Magic" pack, you can either choose becoming a mage or a paladin, and both choices lead to quite a long story...or you can become a roadsweeper, which ends the story in about five minutes.
Kingdom Hearts I and its sequel ask the player to choose which skills to emphasize and de-emphasize (strength, magic or defense) and how quickly they level up at the start of the game. Woe to the player who might unknowingly choose to level up slowly at the start.
In Final Fantasy X-2, at the beginning of Chapter 2, you have to decide which of two factions to give an important MacGuffin to. This affects which faction-related sidequests will be available to you later.
This is also an especially notable example as it also determines whether you can get 100% Completion or not - one path does, the other maxes out at around 97%, no matter what you do. Of course, with the perks of New Game+, full completion is actually somewhere around 145%.
In the Pokémon series, your choice of starter Pokémon determines your rival's starter as well: if you choose the Grass-Type starter, he'll choose the Fire-Type; the Fire-Type starter, the Water-Type; and the Water-Type starter, the Grass-Type (i.e. whichever you choose, he gets the one that's strong against it). Yellow is an exception: there, your starter is always the Electric-Type Pikachu, and your rival's Eevee will evolve into Vaporeon (weak against Electric), Flareon (neutral against Electric), or Jolteon (strong against Electric) depending on how often you lose to him. Your rival's starter (or his starter's final form) will also determine his final team.
Also, in the FireRed and LeafGreen remakes, your choice of starter determines which of the Legendary Beasts from Generation 2 will roam the Kanto region: choosing Charmander will cause Suicune to appear, choosing Bulbasaur will cause Entei to appear, and choosing Squirtle will cause Raikou to appear.
In Pokémon Black and White, your choice of starter Pokemon determines both which gym leader you fight in the first gym (as with your rival, it will always be the one with a type advantage), and also which one of three Pokemon you receive as a gift prior to the battle (which will always have a type advantage over the gym leader).
In Radiant Historia, one of your earliest decisions is whether to stay with Heiss in Special Intelligence, or join your friend Rosch in his new brigade. Due to the nature of the game, of course, the paths are not truly mutually exclusive, and because of some interplay between the two diverging timelines, it is not only possible but necessary to experience both.
Front Mission 3 has quite possibly the most extreme example in all of gaming: an innocuous choice at the beginning (whether or not you want to hang out with an NPC for drinks) determines your entire path through the game. There are two complete storylines with wildly different results throughout, all hinging on that little choice.
Very early on in Soul Hackers, the Player Character is asked what his very jittery companion Hitomi is normally like. The answer given changes one line of her dialog... and Nemissa'sentire skill tree. All the possibilities are about equal in the end, but you're given no indication these two things are related until you answer different next time and wonder why on earth one of your party members is learning wildly different skills.
Real Time Strategy
Battle Realms, at the beginning of Kenji's journey, he must choose weather to slaughter or save peasants. If he saves the peasants the takes the Dragon path, if he kills them he takes the Serpent path.
The very first thing the player must do in Spore is decide whether his character should be a herbivore or carnivore.
In The Nameless Mod, after leaving the starting area, the player has the option to join either PDX or World Corp. It affects the entire plot of the game, most of the missions, and even which locations you encounter.
In the first chapter of Eternal Darkness, you have to pick up one of three artifacts. Which one you take determines which of the three Ancients will be the Big Bad of your playthrough, which also affects which of three attributes (Health, Magick, and Sanity) the protagonists are particularly strong/weak in.
Visual Novels as a genre tend to have this at their core, especially if they have multiple largely independent storylines branching off early in the story, depending on a few innocuous choices early on.
In Tsukihime, a few less than obvious choices during the first two days of the story determine which one of five branches it will follow for the rest of it. It doesn't help that some options and branches only become available after you clear other endings.
Dragon Age: Origins has its eponymous Origin stories: class/race-specific pseudo-tutorial missions, the consequences of which come up again and again throughout the rest of the story.
In Sid Meier’s Pirates!, your choice of nationality and era determines your starting ship, crew, and home port. The era chosen also determines the balance of power among the four nations on the game map. You are also given a choice of one of four different skills, each of which make a different aspect of the game easier to manage.
In Alter A.I.L.A., the choice at the end of the prologue determines which faction you (and by extension Green) will side with for the first chapter, and determines who your allies will be.
Video Game/Fallout3 gives you the option of nuking the town of Megaton fairly early on. If you go through with it, you kill off many characters and lock yourself out of any quests there that you haven't completed yet, which can be quite extensive. If you refuse, you miss out on a free penthouse apartment, which is snazzy-looking but not particularly gameplay-relevant (especially since you can get a functionally equivalent pad in Megaton instead).
Wide Open Sandbox
Spoofed in Saints Row IV, where you're almost immediately thrust into the role of President and asked if you want to sign either a bill for curing cancer or a bill for ending world hunger. In a game that has absolutely nothing to do with either issue.