A very fundamental video game trope, a sidequest (or optional quest, or side mission, etc.) is any part of a video game that is not required to complete the game. Sidequests come in a variety of forms, and completing sidequests generally brings reward to the player such as additional equipment or abilities, areas to explore, supplemental plot related details, or fun unlockables.
Going out of your way and completing all sidequests results in one hundred percent completion.
Some side quests such as the Bonus Dungeon and Bonus Boss may provide challenges more difficult than any content available through the main storyline. This allows more casual players to still complete the game and see the plot resolved, while also giving gamers seeking an additional challenge something to go after.
Tends to be irrelevant to your main quest. Also see Infinity+1 Sword, Easter Egg, Fetch Quest, Quest Giver, Sidequest Sidestory. And That One Sidequest, though you probably don't want to.
Important to note that this trope appears in practically every game which gives the player even a slight amount of free rein, and is one of the best way for a developer to add more content and extend the length of a game.
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Featured in every single Zelda game since the beginning of the series. A well-known example is The Legend of Zelda: Majora's Mask for its sheer number of sidequests, which translates in several characters in need of help, more complex mini-games to play and more secret zones to explore. Other games in the series, such as The Wind Waker, The Minish Cap and Spirit Tracks follow a similar trend. In general terms, the availability of sidequests in a Zelda game is inversely proportional to how many dungeons exist.
In Ōkami, sidequests help Amaterasu to gain Praise units, which gradually enhance her health, paint storage, money and revival chance stats. Some sidequests also house Stray Beads, a bonus supply of Gold Dust (which will permanently power up one weapon) and even secret brush techniques.
Solatorobo has a plethora of sidequests. They're optional, of course, unless you happen to need to be a slightly higher Hunter rank to take a plot-relevant quest. Not all quests will increase your rank, however.
The search for the Extra-Life clover boxes in Little Big Adventure and it's sequel. Little Big Adventure II also has the optional adventure in the form of a Bonus Dungeon; an off-the-main-path island cave off the coast of Desert Island which houses the Protection Spell. You do not need the spell to complete the game, and it will become Lost Forever once you leave Twinsun the second time.
In Stampede Run, you can run down side alleys. They're more challenging (no room for side-to-side motion, tight corners, etc.) but you can get lots of stars if you make it through.
Especially notable is the (only) Face McShooty quest "Shoot This Guy In The Face". Should you find him in the middle of nowhere screaming about how he needs someone to shoot him in the face, accepting means you have to do so. Completing it actually awards you with cash and XP too, as well as the achievement "Well that was easy..."
Most Command & Conquer games features optional objectives in missions, but Tiberian Sun went one step beyond and featured optional missions, unnecessary to progress further in the campaign but granting some sort of advantage in the associated main mission.
Xenosaga Episode 2 was condemned for having what many have argued the vast majority of its gameplay be in the form of crappy Fed Ex sidequests and minigames.
The Elder Scrolls IV: Oblivion has many side quests, several of length and depth to rival the main plotline. Morrowind had much more sidequests, but nowhere near as many as Daggerfall, predecessor to both games, which was 99.9% sidequesting and a bigger game in general.
Though Daggerfall was filled with randomly generated side quests in a (more-or-less) randomly generated game world, while the later games had far more detailed, manually created side quests.
Skyrim and its Radiant Quest system treads on a lot of the same ground as Daggerfall's procedural generation to the same effect.
Baldurs Gate is just full of lazy, lazy gits always asking you to go and fetch them a book, a sword, a dead body, a scroll, or something else that's often less than thirty feet away. To the point where your character has the opportunity to go on a long tirade that anyone who has ever played RPGs will agree with. Your journal will be full of snark about it.
Baldurs Gate 2 is famous (amongst other things) for having no pure Fetch Quests. Subverted and parodied in "Throne of Bhaal", where you subcontract a recovery quest to younger adventurers you just depetrified. They try to kill you for more loot, but reload the game after you slaughter them.
Planescape: Torment has a clever subversion of optional Fetch Quests. An old witch will only teach the PC magic if he fetches her three seemingly-innocuous and useless items: some herbs, rags and a fish. Not only do the apparently pointless errands actually have meaning attached to them but the items themselves are used to create spells.
Paper Mario had a good deal of them as well. The second game made it easier to find the minor ones by means of a Trouble center, where NPCs would put up help requests.
This idea has unfortunately spread to the PSP game Crisis Core. There are 300 missions and just the very first one is required to continue with your game. What's worse is that there are only eight or nine "dungeons" the missions take place in with varying parts of them blocked off and all of them boil down to "find all enemies visible on the map and kill them", meaning the gameplay pretty much requires you do to the same thing over and over and OVER again...
The Last Remnant has a wide variety of sidequests, which is the main way to unlock map locations and the ability to hire some powerful people. There's also Guild Tasks which are similar in function but aren't classed as quests.
Chrono Trigger has six optional sidequests that can be performed between the end of the main quest up to the final boss and the actual fight with that boss. So much Level Grinding is needed to defeat it, though, that they are pretty necessary on your first playthrough, anyway.
Final Fantasy III is essentially full of Broken Bridges and situations which end in benefits for the party (like, say, a nice new airship) and quite a lot of them are just sidequests woven into the main plot, or sidequests that remove something to inhibit you - come on, do heroes normally have to beat up some guy when he chains up their airship?
Similar to Chrono Trigger, the second half of Final Fantasy VI is free form. There are three missions you have to do, but other than that, the missions are all side-quests, with rewards like powerful magic and recovering your warriors.
In Mario & Luigi: Superstar Saga, the Castle Town has two sidequests; you need to find a man's missing 'Beanlets' and dig up ancient artifacts for another man.
Neverwinter Nights. Hey, this Masterson guy in the Docks wants me to find his amulet (and I really wish they'd bothered to mark him on the map). Oh wait, my henchman wants this silver ring I found somewhere. Hmm, I wonder if the other henchmen want anything, like maybe this leaven bread recipe or this weird little brooch. The Many-Starred Cloak people want me to do all this magic stuff for membership and discounts. Oh, I can get money and quest XP for helping with the Tomb of Halueth Never thing. Oh, random if insultingly simplistic escort missions through areas I've already cleared of zombies. Hold on, some random druid wants me to engage in a minor act of ecoterrorism, there could be some dough in it for me. Oh, and my mad rogue skills mean I've been employed to burgle these random nobles. Bear in mind that this is a partial summary of the first chapter of four.
The sequel featured many as well, some of which came back to haunt or reward you as the game went on. One notable type was the inverse fetch quests, where you would find an unusual item then locate someone who could make use of it.
Might and Magic tended to have a fair number of sidequests. VI, in a minor twist, made a fair number of them connected to the main story... in ways that you don't find out until the end of the sidequest, and maybe not even then if you miss a single chest or fail to read that letter you found.
Played with in Dreamfall: The Longest Journey. At one point, Zoe must complete one that involves a lot of running around. She comments on this and there's an option to be lazy and skip a step—with its own consequences.
In The Last Story, sidequests range from single-step item deliveries, going through a very long Chain of Deals, to entire playable chapters that are optional.
In Persona 3, Elizabeth, or in Portable, Theo, offers nearly one hundred-fifty different requests throughout the game, some of which need to be unlocked by completing others, not to mention filling the Persona Compendium.
Robopon has a couple in the first game. There's the Brownie quest with Sam, accessible only at 5 o'clock, rescuing Princess Darcy with the Teardrop of Morris, found by fighting Hunter on Cherry Hill, and the Underwater Creature you give Dream Shells to in order to unlock the Underwater Health Spa.
Rune Factory 2 has an optional sidequest composed almost entirely of fetch quests. The townspeople post requests on a message board that you fulfill to win their money and affection. Marvelous and/or Neverland apparently thought that wasn't enough, so Rune Factory 3 adds a mailbox and a message-delivering owl in addition to the message board. (Each one can only have one request fulfilled per day, so a total of three can be done per day if each one has at least one request.)
Resident Evil 2 has an optional room in Umbrella's facility that can only be accessed in Scenario B. However, in order to get to the room, the player character in Scenario A has to unlock the first lock on the door to the room, then the second character has to release the second lock in Scenario B. The room contains three Lickers and a submachine gun. The weapon is a good find if you had the character in Scenario A take the same weapon from the police station weapons locker. You don't have to go to the double locked room to complete the game, but if you want the machine gun or want to get more ammo for it, it's there.
Resident Evil 3 has a minor side quest with a key item in the disused disposal facility. You get a key that has a scannable card on the key ring and it's used to open a door. However, if you take the key to a certain machine, you can insert the card into it to change its ID. The newly written tag can then be used a bit later on to open a weapons locker that contains a rocket launcher. You can still defeat the Final Boss without the rocket launcher, but having it makes the fight easier.
Turn Based Strategy
Final Fantasy Tactics Advance and its sequel both have tons of sidequests (out of 300 missions, around 30 are mandatory to see the ending). The sequel has up to 400 missions if you count random encounters etc.
The Grand Theft Auto games all feature numerous side-missions that are not essential for completion, but often give you abilities that will make the game easier, such as fireproofing (for completing the firefighting mission) or the ability to get out of jail for free (for completing the vigilante mission).
The Simpsons: Hit & Run has collector cards that contain items from previous episodes (Such as crab juice), collecting all in a level unlocked a multi player bonus track that was specific to the level, collecting all in the game allowed you (in level 3) to trade all of them (although they're still there in the pause menu) for a ticket to the "Itchy and Scratchy: 300 Yard Gash" from the Comic Book Guy.
Non-video game examples:
Otakon LARP naturally has a plot that's supposed to string together the events of the weekend, but you wouldn't necessarily know it from the individual players inventing their own side goals and quests. "Player Specialists" often run a storyline that runs parallel to the GM plot, making what looked like a side quest part of the overall story.
Adventurers!: Karn is obsessed with these, defending them with "Sidequests are an efficient way to increase experience", and once admonishing Ardam that he needs to get his priorities straight when he complains about the team going on sidequests instead of saving the world.
Dagoth Ur: What the hell are you doing here? I'm the end boss!
Ren: Correction, you were the end boss. That's my job now...
Dagoth Ur: But you can't just skip to the end of the game without doing any of the 400 side missions!?
Ren: Fine then, you can go tramping around this God forsaken rock for years on end, doing odd jobs for complete strangers, which in some twisted way ends up saving the whole world for no reason at all...