While Animorphs fandom has a tendency to pair off all the characters, Marco gets it more than any other. I've seen him paired with every single Animorph plus some side characters. (And if Ax is paired with anyone at all, it's generally Marco. Unless it's Tobias, which is squick, for severalreasons.)
Caspian has the distinction of being pretty much the only main-ish character who's not a Pevensie, which makes him poplar among shippers who aren't fond of incest. Ships include Caspian/Susan, Caspian/Lucy, Caspian/Peter, Caspian/Edmund...
Daenerys Targaryen fills this role. Common pairings include Jon/Dany, Viserys/Dany, Drogo/Dany, Daario/Dany, Jorah/Dany, Tyrion/Dany, Dany/Jon/Tyrion, Dany/Aegon and Dany/any-of-her-handmaidens. To be fair, most of those have basis in canon. Viserys feels Dany up at least once, and their family has a good old tradition of incest, while one of the handmaids (Irri) takes it upon herself to sexually service Dany when she's having a restless night. There are several foreshadows (Valyrian polygamy and the three-headed dragon) indicating that Dany could take two husbands in the future (although and she would find Tyrion completely repulsive). As for Khal Drogo, well...
Aegon comes in a close second, with possible matches including Dany, Sansa, Arianne Martell, Margaery Tyrell, or even Arya. As a young man with a solid claim to the Iron Throne, Aegon is easily Westeros' most eligible bachelor.
Hermione Granger has been paired with characters of every age, allegiance, gender, and other realities and worlds. Thanks to carrying a Time Turner around for a year, there are even stories where Hermione is shipped with temporally displaced copies of herself!
Draco Malfoy has been shipped with anyone that is mention he has had eye contact with in the series.
Bella Swan from Twilight is a huge example of this. There are pairings of her with practically every character, including those of lesser importance. And the crossover ones, of course. There's even Bella/Tyler's Van
Percy Jackson has been shipped with every character in his titular series, including very minor characters. Especially since Word of God says that even though demigods' parents may be related, demigods themselves, even if they share the same divine parent, can fall in love and have kidswith no issues.
Luke Castellan is like this as well. He's often nicknamed the fandom bicycle.
The Left Behindhatedom, in their exquisite Bile Fascination, has chosen Nicolae Carpathia for this role. This comes most directly from Fred Clark's famous weekly blog detailing just how bad these books are, the comments section for which has officially branded Carpathia to have canon chemistry with EVERYONE — and they've set out to prove it too!
Frodo Baggins. Especially in the wake of the 2001-3 films, Frodo got paired with every conceivable hobbit, many big strong Men, and an Elf from time to time. Oh, and Gollum. Most of this was generated by women who found Elijah Wood's portrayal seriously attractive. Then again, he is the protagonist.
Legolas, oh so much. Because he is pretty and an elf, and Fan-Girl-Authors really like those things. Legolas/Aragorn, Legolas/Every male elf (including his father Thranduil), Legolas/Gimli, Legolas/Faramir, Legolas/Most of the self-insert characters, Legolas/Sauron, Legolas/Gandalf, and so on and so forth.
In the Rebellion and New Republic eras, it's Luke. It would be Han (because he's sexy), but everyone likes the Han/Leia ship too much. This doesn't stop Luke/Han slash, though it's overshadowed in the fandom by Anakin/Obi-Wan.
In the New Jedi Order era, it's Jacen and Jaina. Not together. Usually. (They're also sometimes paired, or even OT3, with Anakin.) But Jacen and Jaina are paired with anybody who isn't related to them.
Quinlan Vos is this in the Republic comics (though usually he's shipped with Mace, Obi-Wan or Aayla).
The GONE series by Michael Grant has a huge ensemble cast, but for the most part it's averted this trope by the main characters only tending to have one canon love interest... And then there was Lana Arwen Lazar. She's been paired with him (and fanned with by big groups too);
Sanjit, her official boyfriend and a manic pixie dream boy whose only significance to the plot is "fixing" her.
Quinn, with who she has many a UST moment with and there's still loads of sexual attraction between the two.
Cookie, who she used her power to heal in the first book and now he literally only lives to serve her. (kinky).
She's been shipped with the gaiaphage, and unfortunately, due to their disturbing "bonding time" spent together in the first two books, it can be considered canon.
She was even paired with Edilio in the first book, although he came out as gay in FEAR, which made Lana...A beard?
Seriously, the only other character with multiple ships is Brianna, who has two canon ships (with Computer Jack and Dekka), although some argue Dekka doesn't count seeing as it's a one sided crush.
Maximum Ride has Iggy, who in fanfiction has been paired with every member of the Flock (yes, even with 6-year-old Angel), Dylan, and Maya. And that's not even mentioning the sheer amount of Mary Sues and OCs the fandom comes up with. Or even the crossovers...
Protector of the Small's protagonist, Keladry of Mindelan, doesn't have any official love interest. So she's paired mostly with Neal, Dom, and Cleon, the three boys she crushes on... but it's not hard to find fics shipping her with, well, pretty much every character in the series, particularly the Yamani Ladies or other pages/squires.
If the Honor HarringtonWMG page is any indication Shannon Foraker is an extremely popular shipping target. Also doubles as a great pun given her position as the commanding officer of one of Haven's biggest shipyards.