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There are shippings including characters who don't exist in canon, but have the potential to exist, like Portugal, Israel, and Mexico.
England and Prussia have gained a bit of a following as a complementary pairing to the main Prussia/... or England/... pairing. This is particularly odd as neither interact in canon, and have very little history together but is probably a case of being Lonely Together and having large egos.
Canada, being the Ensemble Darkhorse that he is, gets this quite a bit. Canada/Prussia is based off a single piece of dialogue involving an April Fool's joke and maple syrup, and Canada/Ukraine and Canada/Holland are popular due to the aforementioned history classes, even though he has yet to receive a strip with either one of them.
There was this Canada/Turkey one and a Canada/S Italy one. In real life, Canada mostly has a cordial relationship with them, though nothing entirely specific like America or England, and in the manga, has never even interacted or spoken to either of them.
Russia/America was one of the most popular ships in the Western fandom even when they had minimal interaction and zero Ho Yay moments.
Japan gets this a lot in the Japanese fandom as the main Launcher of a Thousand Ships there. Japan/Taiwan is fairly popular because of real-life history and the popular fanon theory that Taiwan was engaged/married to Japan during her stay at his place, even though Taiwan's canon appearances so far have been exclusively with China. Japan/Prussia is even more popular, even though they don't have much real-life history together and haven't interacted even once in the strips. There are a lot more examples, but these two stand out the most for being in the top five pairings for Japan on Pixiv, among the more obvious ships like Japan/England, Japan/Greece, Japan/America or Japan/China.
Unless you include "R-18" in Pixiv search, as this troper tried just now.
There is a little more canon support for Japan/Taiwan now, since the beginning of the "Ilha Formosa" strip has Taiwan cheerfully trying to read a not really thrilled Japan's fortune. Then Hetaween 2011 has Taiwan choosing her costume (as the Japanese virtual idol Silverlight) so Japan could see her on it. Whether that falls in Ship Tease or not is largely in the eye of the beholder.
And then there's America/Belarus, based on a brief note in the manga about how America took care of Belarus when she left Russia's house for a little while. Other than that, the two have never had any shown interaction. Some people do treat America/Belarus as a Crack Pairing, but others ship them seriously, whether because they prefer het to slash or find the potential clash between America's Love Freakness and Belarus's Yandereness amusing/intriguing or the Japanese fanart for them too pretty to resist, or simply want Bela to be out of the way so they can pair up Russia with others without falling in Die for Our Ship.
In a similiar vein, you have the America/Ukraine shippers, who are fewer in numbers. But they do have their reasons, like America's Hero complex trying to help Ukraine whenever she starts crying, his love of her huge... Tracts of Land, and the simple idea that they're almost a romantic Moe Couplet.
Prussia/Liechtenstein seems to have a few followers, and they've never had any interaction. Really, anytime Liechtenstein is paired with anyone who isn't Switzerland; She's rarely been shown interacting with anyone else except him, England (a single strip) or Belarus (some 2009 sketches).
Estonia/Ukraine, too. There's only been a single tiny doodle of them made by Himaruya, paired with some info on them being both in the Gakuen universe's choir club. J-Fen has taken to shipping them quite a bit.
Again, the fourth manga volume caught up on this. Some parts of the Gakuen-based strip focus in the choir club, and it does show them interacting.
France/Germany also has what some might consider a surprisingly large base considering the fact that France and Germany aren't seen interacting that often, and taking into account the time and setting of the main story, obviously not positively (though of course APH is APH). The ship seems to stem largely from their modern day interactions, particularly their involvement in the EU. The G8 strip, which acknowledges this with a theatrically hurt France arguing with the seemingly-henpecked Germany in a amusing manner, was written and drawn after the shipping base had been firmly established, however.
America/Romano is popular enough to have a fairly active livejournal community even though the two have shared little to no screen time together.
Latvia and Liechtenstein have had no screen time together, but it appears that, for some, the adorableness of this Moe Couplet was too great to resist. They have a deviantART club.
England/Belarus is also gaining popularity in the fandom, especially on deviantART and Pixiv. Actually, for what is essentially a complete crack pairing, some of the art on there is very high quality to say the least.
When not being shipped with Hong Kong or China, S. Korea is shipped with Belarus. The two have never even met each other and the only thing that they have in common is the fact that they're obssesed with their older brothers.
Hong Kong/Iceland have had only one or two panels of interaction in Hetaween 2011, and that is it. The thing is, they were a pairing before that. Well, a crack pairing to be exact, but a pretty damn popular one. Especially when you consider Denmark/Norway and Sweden/Finland to be the main pairings in the Nordic 5, leaving poor little Iceland all alone, it gives him someone to be with so that he's not a cock block to Denmark/Norway.
Thailand/Russia may qualify to be one, as both have never met up once in canon. Fans tend to base their interaction on their mostly good relationship in Real Life. On the bright side, this is one of the few pairings where Russia isn't a Psychopathic Man Child, and not too "stereotypical" as well.