Final Fantasy V: Because the game has no Official Couple, Bartz gets shipped with everyone and everyone gets shipped with Bartz. The most popular seems to be Bartz/Faris, but really, he does get shipped with everyone. Naturally, Dissidia has added pairings.
Final Fantasy VI fans can throw Edgar together with anyone and everyone despite the fact that the poor guy couldn't get a date in-game, though he does have the best pick-up line in video game history.
In the Final Fantasy VII fandom, you've got Cloud, Tifa, Sephiroth and Vincent. If one adds in the endless flood of Relationship Sues, a thousand ships is a very lowball estimate of Sephiroth and Vincent's fleets.
In Final Fantasy VIII, Squall seems to be pretty paired with everyone who isn'tRinoa. It's not like the game's entire plot revolves around them falling in love with each other or anything.
In Final Fantasy XIII, we have Lightning. Two major reasons for this are the fact that she has no canon love interest, and the game has a not-inconsiderable amount of Les Yay between her and Fang. Then there's her "motherly" relationship with Hope. If you want to go into Crossover Ships, there's Noctis, and being a Distaff Counterpart of Cloud Strife you can pair her up with him or any of his love interests.
She's also paired with the other party members quite often. Snow X Lightning usually interprets their clashing personalities and Lightning's initial hostility as Belligerent Sexual Tension and takes it from there. Some Sahz X Lightning pair start with them talking about their parental experiences. Oh, and there is some Lightning X Vanille too. She barely talks to Vanille ingame. Her crossover pairings more than doubled after Dissidia: Final Fantasy.
Following somewhat close behind her, there's Squall, who for some reason seems to be paired a lot with several characters.
And in Japanese fandom, the one who stands head and shoulders above ALL of them is Firion, for some inexplicable reason. He is paired with everyone. It's the Rose. That rose has been in the hands of at least nine characters throughout the course of the two games. In fanshipping land this automatically means he is paired with everyone who has obtained it. People even ship him with Aerith for their mutual love of flowers, even though the two have never canonically met! However perhaps the only pairing that has any real basis in the events of the games is Lightning, due to their undeniably cute scene in the reports(turns out he just wants that pesky flower back but still who wants to crush a shipper's dreams!)
Lightning's paired with a fair amount of men in Dissidia's sequel (technically prequel), too. Mainly, Kain Highwind, Firion, and the Warrior of Light.
The World Ends with You is still a new fandom, but already Neku's being paired with everyone. Minamimoto/Neku? Where'd that come from?
Minamimoto: Why, hello there! This mic is zetta sexy!
Also calculators and math books.
Joshua too. Although Neku is by far his most popular option, he regularly gets shipped with Hanekoma, Minamimoto, Kitaniji, and Rhyme. Shiki and Beat are rarer but also exist, and he's a favorite for crossover ships.
Yu Narukami of Persona 4 has been paired with every conceivable cast member of the series at one point or another.
Somewhat justified since both games included building relationships as a major part of gameplay, both games do allow for romantic developments with characters of the opposite sexnote In 4, prior to the Updated Re Release, Yu could even get something of a harem going on, while the Updated Re Release for 3 added the option to play as a female protagonist who could get it on with all the boys, and 4 includes a male ally strongly implied to be at least bisexual.
Rune Factory 2 has an officialLauncher of a Thousand Ships candidate in the form of Yue. When the main character marries a girl who was destined for another guy, that guy will automatically end up with Yue.
He has been paired with his own father. Granted it's usually before The Reveal, but some even have the relationship continue afterward.
Because of the Multiple Endings, Lloyd is a canonical Launcher of a Thousand Ships.
It even continues in the sequel! Dawn of the New World confirms that Lloyd scores with whoever you want him to.
Emil seems to be this, too—he's been paired with a good chunk of the new cast including Ratatosk and Aster (who Ratatosk killed about a year and a half before Emil even came into existence.) Plus several of the original cast.
As far as Tales of the Abyss goes, Luke/Everyone might as well be canon. ... Well, everyone important. At the very least you cannot deny that Luke/The Party (or The Party/Luke, as it is...) is very nearly canon.
Guy gets this treatment too.
Jade also has his fair share.
If you are a Tales of Vesperia character, the fandom has decreed that you want to have sex with YuriLowell. Period. He's probably most often shipped with Flynn, but Estelle and Judith aren't far behind...and then there are people who ship him with Raven or Duke or Rita or even Ioder, and, appropriately enough given the above, Yuri/Luke is a popular crossover pairing.
Tales of Xillia 2 has two, in both Ludger, the new silent protagonist, and Jude, whose Ship Tease has gone Up to Eleven since the last game. They get shipped with each other, of course, but Jude also gets shipped with Alvin, Milla, Ivar, Gaius (thanks, Hot Springs ending), has a canonical alternate-universe-bird-spirit boyfriend (it's complicated) and even gets some Ship Tease with Elize in a victory quote. Ludger gets shipped with Alt!Milla, his brother, Nova, and...pretty much every member of the playable party since they've all got relationship values now. It's madness.
Fire Emblem has Roy. He has 6 possible wives, subtext with two girls he can't support, and can support with 4 other males, one of which is his milk brother (or male maid), and is obviously a target for Ho Yay/Bodyguard Crush. AND he was playable in Smash Bros. Melee...
Roy's got some brutal competition for the "Pimp Lord of Fire Emblem" from Celice in Genealogy—Celice can take to his side one of 15 possible wives (7 female children and 7 replacements, and then Yuria).
Lyndis from Fire Emblem 7 already has 4 canon love interests (5 if you count Florina as one), Sain and Wil are considered extremely plausible, Karel/Lyn and Lyn/Ninian have niche followings in the fandom, and one author even managed to pair her with Renault and make it plausible. There's the infamous Lyn/Tactician fics.
My Unit (or Chris) in the remake of Mystery of the Emblem is an in-universe Launcher, capable of supporting with the entire cast of the game. They don't have as many potential love interests as Roy or Celice, though; male MU has 3, and female MU has 5.
The My Unit (or Avatar as s/he's called in the west) from Fire Emblem Awakening takes this one step further. S/he can marry any player character (including the Paralogue characters Gangrel, Walhart, Aversa, Emmeryn, and Ike's descendant Priam and the Child characters like Lucina, Nah, or Yarne... and yes, s/he can have a kid with them), provided they're of the opposite gender.
Heck, the only Awakening characters this isn't canon for are ones like Say'ri, Tiki, Anna and the Spot Pass characters. And even that doesn't stop fans from coming up with their own fanon ships.
Link is easily the video game town bicycle; he's paired with everyone from the eponymous Princess Zelda to... well, to the Shinto goddess Amaterasu as seen in Ōkami, given that both were wolves recently. (If there were a Shinto Hell, whoever first did that would be going there. Nor is this the only strange ship involving Amaterasu, in or out of Okami itself.) Link gains a new harem within each big game. Ocarina of Time Harem,◊ Wind Waker Harem,◊ Twilight Princess Harem,◊ and much much more!
In Golden Sun, everyone is a Launcher of a Thousand Ships.
And then the Lighthouses are jokingly considered as such in both the phrase's literal and other metaphorical sense, thanks to the fact that more than a half of the non-crack pairings (and a few crack ones) get their biggest Ship Tease in them.
World of Warcraft fan fiction's poster girl is Lady Jaina Proudmoore by far. She's often shipped with Arthas (pre- and post-undeath), Warchief Thrall (a pairing once characterized on the official Blizzard WoW General Discussion Forum as "Bow-Chikca-Lok'Tar"), a triangle between the three (and I'm sure a ménage ŕ trois has been done at least once)...every fan fic writer who writes about his own PCs will have them meet Jaina at some point.
Jaina/Arthas (pre-undeath) actually is canon: they used to date when they were both teenagers.
And yes, they did sleep together. They would have married too, if not Arthas decided they both should finish their studies first. And when they were done with that, there was that whole undeath plague...
Kael'thas Sunstrider! He canonically had a crush on her (although it was apparently unrequited).
Jaina was heiress to the throne of Kul Tiras before the fall of Lordaeron, and Arthas and Kael'thas were both crown princes of their respective allied kingdoms. Given how royalty is often married off to allied royalty, its not surprising that she'd have been socially introduced to every male heir in her age group from every ally Kul Tiras had.
Thrall and Sylvanas fit this trope for Horde aficionados.
Outside of canon characters, elves in general seem to fill this role. Especiallyblood elves.
And as of Patch 5.1 and the Tides of War tie-in novel, Jaina is canonically hooked up with Kalecgos, who is literally a dragon.
Rinnosuke Morichika, being The One Guy, seems to get this treatment by default unless he's being portrayed as the Memetic Molester. If you scroll down this page, you will see a chart that breaks down his ships with every character in the series by source material.
Though specially Marisa, with more than seven frequent pairings, is who stands out in this trope. In some works to Memetic Sex God and GARlevels.
Yuuka has quite a few ships too.
Metal Gear's Snake tends to be paired up with almost everyone in the series, and outside of it - even before his appearance in Brawl. It really doesn't help that he's the Prince of the Even the Guys Want Him trope (Big Boss being the king).
In the MGS fandom itself, Otacon's probably earned this. He's been paired with Snake, Mei Ling, Raiden, Liquid, Grey Fox, Big Boss, Mantis, and young Ocelot, time paradoxes and plausibility be damned.
Sonic and Tails in the Sonic the Hedgehog series go with everyone... even though they show the least romantic interest of the entire cast.
Disgaea has a rather small shipping base, in that most of its characters are only commonly shipped in one or two pairings each... the second pairing usually being with Etna. Seriously, Etna gets paired with everybody, including but not limited to: Laharl, Flonne, Mid-Boss, Adell, Rozalin, Axel... even the Prinnies.
Team Fortress 2 - The poor Scout doesn't get a break when paired up with any of the other classes, not even his own. There's also the Pyro, who may be female. The Medic (usually a Memetic Molester) occasionally gets this kind of characterization due to his status as a Support class and the fact that he racks up points the fastest via assisting other classes, as well as the fact that his gameplay role is that of the White Mage and therefore Yaoi Fangirls designate him as the de facto Uke when Scout's not filling that role.
Poor, poor Splash Woman. It doesn't help that she's one of the only two lady bots in a cast nearing 100.
Looking at the fan art, Metal Man gets this treatment too.
In general, ANY Robot Master, including the Rockman Killers and the Stardroids. Each of them have launched enough ships to make a tangled love web. Perhaps the most common: little Mega Man himself!
Like their brother, Roll and Protoman also have launched their own fleets. Protoman is especially popular judging by the sheer volume of fan art and fiction that exists. In Roll's case, she's gets paired often so often for the same reason Splash Woman does.
Leon Kennedy, of Resident Evil is a classical Type A, sometimes even to the point of Even the Guys Want Him among players. His canon love interest is Ada (and to a lesser extent, Angela, who appeared once and then was Put on a Bus), but he's also regularly paired with Ashley, Claire, Jill, Steve, Chris, Wesker, Luis, Krauser, Hunnigan, Kevin, Manuela, Buddy, Helena, Sherry, and Mike the Helicopter Guy. In addition to the large amounts of zombie porn that affect all characters within the fandom and countless self-inserts. Admittedly, the fact that he almost always ends up with a female character for a partner and the fact that at least two of them seem to have crushes on him does not help, nor does his (failed) attempt to get Hunnigan's number.
Claire Redfield, however, falls quite neatly under Type B as the "kickass everygirl" who has no special training and a kind heart, but still keeps up with the rest of the characters. Her ships includes Steve, Leon, Ada, Sherry, Alfred, Wesker, Jill, Rebecca, Carla, and her brother Chris, though many crack pairings exist for whatever character the author deems most attractive.
As mentioned above in the discussion on Link, Ōkami's Amaterasu gets shipped with far, far more characters than a non-talking wolf should ever be. Most of these rely on Ammy having a human form that we never see.
This happened to Dawn and Lyra from Pokémon. EspeciallyLyra. The most popular characters to pair Lyra with (in Japan, at least) are Silver, Lance, and PETREL. These are just the popular ones.
In the west (and East at that), Morty and Proton are also often shipped with her.
Silver himself is a shipping magnet, too, especially when you factor in Special.
With Pokemon Black And White, we have the male protagonist Hilbert (more commonly referred to as Touya or Black), who's being paired with practically every female in the game.
There's a reason a why a lot of players name their Ditto "pimp".
Haseo of .hack//G.U., poor guy, virtually has a harbor reserved for him. From Atoli to Shino to Ovan to Endrance to Skeith to even the Grunties, he's been stuck with EVERYONE. Find someone that he HASN'T been shipped with.
Like Lloyd in Tales of Symphonia, Haseo is an actual canon launcher, since at the end of the third game, you can choose to "marry" (the characters are playing an an MMORPG) any party member, regardless of gender or even age. When trying to do it with a man, usually it leads to a "best buddy" scene rather than marriage, but there are a few exceptions like Endrance or even Azure Kite who want a little more than being friends...
Chun-Li from the Street Fighter series is also a fairly infamous case of this as well as she has been paired with Charlie, Ryu, Ken, and Guile (and to a lesser extent Cammy).
Ryu gets his share, too, despite his lack of interest in love life, being paired with Chun-Li, Cammy, Sakura, Juri Han, Rose, Ibuki (?), Makoto (??), Ken (oh boy)... and Bison? And maybe Sagat, that is, if he's not busy with his cornflakes.
Ellis from Left 4 Dead 2 ends up with everyone. Although 90% of the time it's with Nick, you can find him with his canon love interest Zoey, teammate Rochelle, Coach, the Hunter, the Witch, the Smoker and his friends that don't even appear in the game like Keith. Ellis will also usually be the target of any fanfic writer's Original Character.
Kasumi from Dead or Alive has ships for every base, ranging from the fairly mainstream Ryu Hayabusa, the more yurilicious Hitomi or Christie, all the way to straight up incest with her half sister Ayane or her brother Hayate. It would seem that whoever the favorite character is, shippers generally feel they go nicely with the cute redhead ninja.
Jin Kisaragi from BlazBlue. Aside from him being coupled with his older brother/rival Ragna the Bloodedge and childhood friend Tsubaki, Jin has been coupled with almost every character from the series even if it's almost impossible to happen. Despite him being a total asshole, this must be just a huge turn-on for the fandom.
Ragna himself enjoys a fairly substantial shipping base as well, having, in addition to Jin, been paired with Noel, Rachel, Taokaka, Nu (kinda-sorta), Celica, and several characters who aren't even from the same franchise.
Carl Clover also enjoys a fair amount of popularity amongst shippers as well, ranging from Noel and Makoto, to Litchi and his own sister, Ada, and, like Ragna, with several characters who aren't even from the same franchise.
Advance Wars has three. You have Eagle who gets all the good guys, Hawke who gets all the bad guys and some of the good guys, and Andy who gets it up the butt from everyone. ESPECIALLY Eagle and Hawke.
Reptile seems to be working his way up, too. Although the only character that shows romantic interest in him is probably Khameleon (due to the whole Adam and Eve Plot thing), he's additionally paired with Mileena and Nitara on a somewhat regular basis. Apparently, the lizard man has moxie.
Chell has been paired up with nearly every character in Portal. Wheatley is paired with almost as many. And Chell/Wheatley is the most popular ship in the fandom.
As the only female character in Live A Live who can be brought to the final battle, and hence the only one who can canonically meet the protagonists from other eras, Li has been shipped across time and space. Popular love interests include Akira and Odio, but Masaru and Oboromaru are also occasional fic subjects, as is Yuan in the Fix Fics where they both survive.
Speaking of Super Smash Bros., Samus Aran easily falls into this due to not having any romantic interest in her own series (or more than a handful of characters per game other than herself, for that matter). Master Chief, Boba Fett, and Iron Man are common choices due to Shared concepts, but she seems to pick a new one up every new game in the Smash Bros series: Captain Falcon in Melee, Solid Snake in Brawl, and now Little Mac, due to the latter's introductary trailer, even though it primarily involved him punching the hell outta her for making fun of his height.
Also, oddly enough, Pit and Mega Man in the latter two games as well. We're not sure why parts of the fandom have deemed her a shotacon, out of all the characters.