Patamon and Gatomon are arguably the most popular and/or mainstream Digimon couple in the series, that includes the entire series till season Six where Digimon romances are actually relevant to the plot. In addition they seem to have a special bond in nearly all their forms. Of course like a lot of Digimon couples it could be a case of Scotoma (the eye seeing what it wants to believe).
Patamon and Gatomon
Also known as PaGato by the fans. Much like their human partners they are almost always seen together, especially in season two. Though it is rare that they show any significant signs of affection to each-other, it is clear that they are close friends. One scene that can be depicted as romantic is season 2 episode 22 when they are dancing together at the beginning.
Angemon and Angewomon
What could be considered an even more popular or at least more mainstream couple. The two of them are seen together quite a lot and have shown to work well together. The two of them were working particularly well together in both Digimon 02 movies, their very presence could be a result of fandom. In the english version of Revenge of Diaboromon while racing across the web to fight said Diaboromon they briefly argue about directions in a manner similar to the old married couple cliche. In season 2 episode 7 Patamon boasts to Davis that Angemon and Angewomon "make a great team", not necessarily romantic but the fans (and Davis) seemed to interpret it as such. In fact the whole scene could very well be a nod to their potential couplehood.
(Side note: Davis asks if there is a baby version of Angemon, likely as a gag from the writers till season three where MarineAngemon makes and appearance which could be considered a baby form of Angemon).A stronger sign of romance appeared during the Dark Masters ark in the first season of Digimon. When Machinedramon was defeated and falling to pieces Angemon was briefly seen with his arm wrapped around Angewomon shoulder both looking happy with the position. Another scene is episode 50 before engaging in a cat fight with LadyDevimon Angemon leaves to get back-up as he does Angewomon smiles warmly to Angemon.
Pegasusmon and NefertimonThe two of them were also seen working together many times, though it could be based more on practicality then any significant feelings. Never the less the fact that they have a Unison attack Golden Noose is evidence of a special bond between the two of them.