Super-secret character mentioned directly in the manual and whose weapon of choice is conspicuously for sale before you have a party member who can use it. They aren't required to stay in the party, but may temporarily join you. If you really want them back, some games will let you hunt them down again, often by some non-intuitive and convoluted means.
The biggest problem with these characters is getting them can happen very late in the game. Although most games are kind enough have them near the same level as your characters, special skills or weapons that need to be built up are going to take a long time to get. In addition, since they're optional, the majority of the plot can't assume you have them. Thus they'll have less impact.
Similar to the Guest Star Party Member, the Optional Party Member is commonly either hacked into the party or used as part of a challenge. Sometimes they may be a Joke Character who isn't very useful but other times they may be one of the best characters in the game. They might even be both.
Many Strategy RPGs will use this trope a lot, as in fact the story will often be written with the thought of how the party member may be permanently killed off at any moment in the story. Most of the time it is when they are not mandatory, but there are exceptions.
Compare Guest Star Party Member.
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In Super Smash Bros. Brawl's Subspace Emissary, every playable character in the game joins throughout the story, except for Wolf, Jigglypuff, and Toon Link, all of whom, like everyone, can be unlocked outside of Emissary by other means. To use them in Subspace Emissary, hidden doors must be found in three levels.
In fact, by the time you clear the "Entrance to Subspace" stage, everyone besides King Dedede, Luigi, Ness, Kirby, Sonic, and Bowser becomes optional party members — you just need to collect their throphies during the subsequent two levels (Ganondorf, however, requires both Link and Zelda to be unlocked). And at the beginning of the game, Kirby has the choice of having either Peach or Zelda accompany him until "The Lake Shore" stage.
Hack And Slash
To beat the game, you don't need to collect any party members in Drakengard. To get the fourth or fifth ending, however, you have to have all of them. This causes some problems for the sequel, which is based off the first ending and has a former party member appear in a position of power eighteen years down the line, even though technically you didn't have to get him in the first game.
Mega Man X 8 allows the player to unlock three extra playable characters, the Bridge Bunnies Alia, Layer and Pallette (Distaff Counterpart to the three established heroes X, Zero and Axl, respectively); for them to be available, you must use an individual Navigator several times more than the other two in one playthrough, then buy her in R&D in a New Game+.
In Puzzle Quest: Challenge Of The Warlords, Only three of your potential nine party members are mandatory - Darkhunter, Khalkus, and Sunspear. You'll meet Flicker, Winter, and Elistara along the course of the story and can recruit them via side-missions. Princess Seraphine, Drong, and Patch require side-missions to find at all. If you follow the Plague Lord quest from the iPhone version or the Plague Lords expansion, Galnoth the Dark Elf will join you (or rather, you'll force him to join you). You're also not obliged to have anyone actually in your party for the vast majority of the time, and if you make the wrong choices you can get Darkhunter to abandon you forever near the end.
In fact, all of your good-aligned party members will leave if you release the necromancer, then follow the path suggested by Bane.
Real Time Strategy
WarCraft III, special campaign in the "the Frozen Throne" expansion. The very Diablo-like campaign is centered on half-ogre Rexxar who may be joined by other characters, and one option is to include Chen Stormstout, a humanoid panda, a kind-of Easter Egg character. Since he's optional, he doesn't have a single line of dialogue since his recruitment, nor any action specific to him.
There's also a panda appearing in the Blood Elf campaign. He'll be with you for one mission if you beat the secret level.
The OgreBattle: The March of the Black Queen series was excellent in that while most of the party members are optional, most provide interesting dialog and many scenes play out differently with different combinations of NPCs. Even the mandatory party members usually give you the "option" of killing them.
Beyond The Beyond has Tont, Lorelai and Percy. Tont is basically thrown into your lap, and Lorelai is easily missed. Percy actually joins your party early on in the game, but doesn't return until much later under the guise of the Black Knight. To get him to rejoin, you have to actively avoid attacking him when he fights you.
Star Ocean games feature these en masse. Many of them are obscure, but there are also typically more optional party members than you have spaces in your party, requiring players to go through the games multiple times if they want to see everyone.
Nobody in Star Ocean: The Last Hope is optional, but in New Game+ you can choose between Arumat and Faize for who you want to stay in the crew during the midpoint. Faize does not participate in PA past that point, however.
Everybody except for the two primary protagonists in Star Ocean: The Second Story is optional. This had the unfortunate side effects of them having to be absent from all of the animated cutscenes in the remake, and also keeping their dialogue to a minimum outside of sidequests and private actions. Some of them were also mutually exclusive, which makes the fact that all of them show up the direct sequel as mandatory characters a little awkward.
In the third game, you can pick any two of the following: Albel, Nel, Roger,(of these 3 at least 1 has to join you, usually Nel unless you trigger sidequests for the other 2) and Peppita.
It's not unheard of to go the entire game without ever seeing Roger at all. His introduction and event chain kick off at a certain cabin in the woods which is easy to bypass the first time you visit. Complete the quest in the woods without ever visiting that cabin and his appearance will disappear completely. No outside dialogue will ever point to him, and for the rest of the game you'll wonder in mystery who this entire class of unequipable weapons in all the shops you find is meant for.
Mog, Umaro, and Gogo are optional characters in Final Fantasy VI; you also could try to defeat the last half of the game without picking up the majority of the characters you had in the first half. (In fact, it's technically possible with only three of them!)
Mog less so than Umaro or Gogo, since Mog appears and fights with you at the beginning of the game.
You can optionally recruit Shadow at certain points in the first half of the game, but he'd always leave your party sooner or later, sometimes even randomly after a battle. He does join your party for two plot-related events though. In the second half of the game he can only be found again if you waited for him on the Floating Continent. If you did, you can recruit him permanently from that point forward.
Final Fantasy IX: Quina is a permanent party member (s/he officially joins just before Fossil Roo), but is optional before you enter Gizmaluke's Grotto on Disc 1.
And recommended. Gizamaluke is easier with Quina helping you, and he's still a pain even then.
Chrono Trigger has Magus, who starts out as a boss; however, you can choose whether or not to fight him at one point, and if you don't attack, he joins your party. In addition, the main character at one point dies, and it is entirely optional whether you take the sidequest to bring them back. Obviously, these two won't appear in the ending if they're dead.
Robo can also be made optional, as you don't have to get him back after leaving him during the Fiona's Forest sidequest. You have to make a conscious effort not to get him back, but it does mean you can finish the game without having Robo, Crono, or Magus.
Chrono Cross has an impressively long list of optional characters, and each has dialogue parts in his/her own dialect and speaking style, even in the game's ending. With New Game+, it is more than possible to have even certain villains as party members. In fact, you need to play the game at least three times to be able to recruit everyone.
The Suikoden series is king of this trope, with there being 108 characters, only a fraction of whom are required to actually finish the game. About half of these optionals barely differ in appearance to normal townsfolk.
Virtually every NPC in Radiata Stories can be recruited into the party in some manner. You end with a cast rivaling Suikoden in size, filling your party with the entire ranks of every guild, most townsfolk, and a number of monster characters.
However, unlike Suikoden, the only use for these characters was in battle. Combine this with an inability to change the skills and equipment of anyone but the protagonist, and at least 100 of them are just JokeCharacters. Additionally, several party members are mutually exclusive.
Easy in a physical sense, but the endless whining of another recruited character makes it just as easy to avoid them out of spite. Not avoiding Cream to peeve Amy, mind, but avoiding Omega to peeve Shadow. What makes it even worse is that Shadow is supposed to be almost emotionless.
They're worth it though: Cream is a better healer than Tails (and the most important character to equip with the Chao that automates POW moves), while Omega is an offensive powerhouse and talks like HK-47.
Dragon Quest VI has a couple of these. The first one is Amos, who is cursed to become a monster every night; if you can get him a special seed, he can control the transformation and joins you. Additionally, if you have the Beastmaster class, you can actually recruit some of the monsters you come across and make them join you as permanent party members, who all have their own unique ability sets. This is removed in the remake, but in its place, several types of Slimes can be recruited. Finally, in both versions, Lizzie the Hackasaurus is available after Terry is recruited, and unlike the other monster units, she has plot significance and in the remake, proper Party Chat dialog.
A number of characters in Dragon Quest V (capturable monsters, human allies, etc)
Marvel: Ultimate Alliance has several optional party members, but Blade is the oddest by far. He joins the party after you save him from a giant claw vending machine in the Murderworld level, but the real question is why he's there in the first place. He has no dealings with the park's operator Arcade (an X-Men villain) in the comics, and there's no explanation for his kidnapping in the game (he just blacked out and woke up there). Even when you fight Arcade, he makes no comment on kidnapping Blade at all. At least he's fun to use.
Fawkes is required for part of Fallout 3, but soon leaves
He returns to assist you during the destruction of (and your escape from) Raven Rock, though like other companions, he requires a specific karma alignment (Positive).
Most of the party members in Baldur's Gate, though getting rid of Imoen and Jaheira (aside from having them die) is next to impossible, particularly in the first game. In general, though, it's harder to get specific combinations of characters to 'stay' in your party (Minsc and Edwin, for example, or Keldorn and Viconia)—just having them join is easy enough.
Knights of the Old Republic. Most of the party members are mandatory and probably few people actually go through without getting them all, but Juhani can be killed on your first encounter, HK-47 does not need to be purchased and in the sequel, you do not have to repair him. Also in the sequel, there are two pairs of characters who are interchangeable, depending on your gender and alignment.
In the .hack// series of games, following plot points in The World's Message board — especially those related to finding rare quest items — will often reward the player with the poster of said messages as recruitable allies. Examples include Natsume, Sanjuro and Gardenia.
Wild ARMs 2 lets you recruit Marivel, a odd vampire like girl that helps you out on the sidelines through out the main story. She's found in an optional dungeon called the Crimson Palace that can only be reached after getting the Global Airship.
Many of the characters in Dragon Age: Origins can be considered optional. Some can simply be avoided or refused entry into your party. The most notable example is Loghain in that accepting him will actually cost you one of your teammates. Perhaps the most easily missable is Sten - who is tucked away in a cage and freely admits he's a murderer.
In Awakening, you can wind up without any party members by giving Anders to the templars, refusing to let Oghren and Justice join the Wardens, ordering Nathaniel to be hanged, killing Velanna and refusing to let Sigrun accompany you into Kal Hirol.
In Dragon Age II, it's possible to miss Isabela simply by not bothering to go back to the Hanged Man. And if you refuse to help a dwarf get his cargo of illegal lyrium back, which most party members object to, no less, you'll never discover that that secondary quest was actually a ploy to help (and meet) Fenris.
The SEBEC route of Persona only allows you to take one optional party member out of an available four (Brown, Elly, Ayase or Reiji), while the Snow Queen route lets you take two out of three (Brown, Elly and Nanjo). This means you'd have to play the game fourteen times to see all the different characters and interactions in every ending. Hey, they had to put Replay Value in there somehow!
In Mass Effect it's not necessary to recruit both Wrex and Garrus, as the player can proceed with the game after recruiting one and leave the other behind. Similarly, in the sequel the player can progress to the endgame after recruiting Tali, Thane or Samara, skipping the other two characters entirely, and it's possible to either sell Legion to Cerberus and/or not wake Grunt from his tank. That's not counting DLC characters Zaeed and Kasumi.
Of course, if you take that approach, it's highly unlikely that anyone will survive the assault on the Collector Base. You've got enough people for the required specialist slots, but without both Garrus and Grunt, the team's defensive score might not be high enough for everyone to make it out of the final segment alive.
Don't forget Morinth, who joins you only if you let her kill Samara during her loyalty mission.
In the 3rd game you can turn both Tali and Ashley / Kaidan down when they offer to rejoin. Or, alternatively, get them killed. Garrus has to join you if he's alive, but you can do something about that in the second game. You can also pass up Javik by not doing his mission, though as with Kasumi and Zaeed, it's unlikely a player would buy a DLC pack and deliberately not use the content.
The interesting result of all these shenanigans is that there are only three characters who actually can be recruited in all three games, and with careful maneuvering you can actually set it up so that none of them are with you from start to finish. In fact, you can play the series with certain characters simply not appearing at all (Samara, Thane and Wrex are good candidates). And again, this is before DLC kicks in.
Einherjar in Valkyrie Profile 2 Silmeria (not sure about the other games) are entirely optional. Although you do get some stat growth items for leveling them up and releasing them from the party.
The original Valkyrie Profile only has two party members that you must recruit, Arngrim and Jelanda, both of which you get in the prologue. Recruiting anyone past this point is optional. Due to how the game is set up, you need to recruit at least a couple people to send to Valhalla to not get the C-ending, which is effectively a game over for failing your mission. The A-Ending does require you also recruit Mystina and Lucian because both characters are involved in this ending.
There are three in Shin Megami III: Nocturne; Black Frost, Dante, and Samael. Black Frost can be recruited a while after you defeat him as an optional boss, Dante automatically joins you in the Fifth Kalpa (which is entirely optional, but unavoidable if you're going for the True Demon Ending), and Samael will only join you if you chose to side with Shijima.
The Last Remnant union system is made of this trope. You have 49 'unique' leaders, each with their own voice acting and backstory (8 of whom are important to the storyline). On top of this are 118 'ordinary' leaders and over 200 common soldiers that can be hired. Every single one has a unique combination of stats, weapons and fighting styles: all this for a game where you can have a maximum of 6 leaders and 12 soldiers in your party. The main character Rush is the only one who has to stay in the party; everyone else eventually becomes optional.
Pirates of the Caribbean, a game based very loosely on the movies has a field day with this. With the exception of two characters who are mandatory (and also impossible to kill) in the last part of the game, all characters are optional, randomly generated and fully expandable.
Death's Hand from Jade Empire. After he is defeated, the Spirit Monk has the option to bind his soul, thus gaining him as a party member.
Every. EVERY. Pokémon. Well, except your starter. And in Pokémon Black and White, Reshiram/Zekrom, unless you took the extremely tedious route of filling your entire PC before fighting them.
Even after you've caught your second mon, you can retire your starter. The game requires you to have at least one Pokémon in your party at all times (You need at least two in your party if you want to trade in later games).
In Drakensang some characters (like Rhulana, Gladys, Forgrimm and Traldar) are practically vital to the game and can be found along the main quest. Other characters like Dranor, Nasreddin and Jost can be hired only after completing optional sidequests (or, in Nasreddin's case, pay him enough cash). Alternatively, you can get all the possible party members but keep them unused at home in Ferdok.
Turn Based Strategy
Technically, everyone but Ramza is an Optional Party Member in Final Fantasy Tactics as you can kick anyone out of the party at any point and even refuse them when they try to join up. The game has five actual optional characters (Cloud, Beowulf, Reis, Worker 8/Construct 8, and Byblos), and one more (Balthier) in the PSP version.
Super Robot Wars has tons of these in every game, usually people who normally died in their respective show or some way should leave you. Or villains that fans liked or were especially sympathetic. They also have a habit of adding in Gundam mecha or Mecha Expansion Packs that only exist in model form, such as Hajime Katoki's version of Wing Gundam in Alpha 2, or the Strike Gundam/Strike Rouge's IWSP pack.
Likewise, in Battle Moon Wars, you can unlock Sacchin in Takumi's route, kill her with Shiki two out of three times she appears, and Mech-Hisui in Haruna's route, get a combined total of 50 kills with Hisui and Kohaku.
Shining Force II: The Sword of Hajya had a small handful of these, one of which was hidden in the most obscure possible place: the third bottom pillar on the right of the entrance to the throne room in a specific mission. You attempt to talk to the pillar, and out pops, not a ninja (that would make too much sense), but a samurai. Seriously.
The Shining Series does this a lot. In Shining Force III: Scenario 1 for example they do the same ninja trick, twice in Part 1. If you're Japanese and lucky enough to own all three parts of the game the decisions you make in the other parts affects which characters you gain in all parts, which becomes a massive mess of carefully setting it up in one part just encase you want them in the next. If you're not Japanese you can still go to all the trouble but gain none of the reward.
Again in Shining The Holy Ark Doyle, the kick-ass Werewolf Ninja that helped you out and stalked you at the beginning of the game can be found pretending to be a tree in the first village near the middle of the game.
Let's also not forget Sheena hiding behind a waterfall in Shining Force II.
A significant portion of the later characters in the original Shining Force game.
Due to how Fire Emblem games work, all but a handful of cast have no dialog after they are introduced. They can die forever without affecting the plot, and you can miss them entirely without noticing.
However, the only time where that character is obviously missing is in Fire Emblem: Radiant Dawn: Pelleas isn't available first playthrough, however he's the only playable character that wields Dark Magic. So you can have multiple Dark Magic tomes without anyone to use them.
Not the only one; Lehran can also use them. He's not available in the first playthrough, either.
This is more Egregious than what one may think, even for Fire Emblem; while it's fairly simple to get Pelleas for your Dark Magic needs, getting the extremely powerful Lehran is Guide Dang It at its finest. It doesn't help that the most important requirement is basically a Luck-Based Mission that practically takes up a third of the game. Another character that is also Guide Dang It is Stefan, but at least he's not hard so much as annoying to get.
Also, not really the only time. In FE 7, your only dark-magic user, Canas, joins in one of the Gaiden chapters. Granted, you do get multipurpose Athos right at the end, but you pick up a lot of Dark magic before that.
Similarly, in Fire Emblem: Shadow Dragon (a remake of the first game in the series) the only character who is a dark mage by default is available only in a Gaiden chapter more than halfway through the game (said chapter requires you to kill off most playble characters to access. Of course, in this game, most characters can switch classes, but almost no one one else is actually designed to be a dark mage.
Not that it really matters since magic isn't really specialized in Shadow Dragon beyond the point of "tomes" and "staves".
And normal mages are better anyway, as they tend to get much better growths.
Many party members in Fire Emblem can in fact even be missed. Some of them have to be recruited by talking to them (And won't attack) or they have to be found in a village.
In Path of Radiance, it's possible to turn down Heron Prince Reyson's offer to join you. You'd have to be a total idiot to do so however, as not only is Reyson incredibly useful, but you also miss out on two other characters (his bodyguards) AND you're forced to fight an incredibly tough boss in the next chapter. (Reyson can talk him out of fighting you) All you get out of refusing him is a Renewal scroll, which is practically useless. (And in the Japanese version, it actually was useless, as the only two characters compatible with it already knew the skill.)
Fire Emblem Awakening takes this to extremes, with the majority of the cast being technically optional. Donnel and Anna are only found in optional missions, and just completing them is no guarantee you'll recruit them. Gaius, Tharja and Libra can all be killed off in battle before you can recruit them. Then there's the children characters who are recruited through side missions you may not even unlock. And then there's the Spot Pass characters who only appear in optional missions near the end of the game if you donwload certain content.
There are also a number of characters from previous games who can be recruited via Spot Pass, though they have no lines, no support and you have to recruit them with your hard earned G.