Game of Thrones is probably not the series that springs to mind when talking about successful romantic relationships - it's hard to find any that don't end in murder and misery, although that does put the average breakup into perspective! However, that doesn't mean that every relationship in the series is flat-out awful (even if some do set a new bar for doomed romance).
In fact, a few relationships actually manage to be fairly successful, based on the people involved getting what they want, staying together for years, or maybe just being a good example of a healthy and strong love. Admittedly, it's possible to count on one hand the number of romantic pairings that manage all three of those measures of 'success', but some are definitely much, much better than others.
Least: Sansa and Ramsey
This should be obvious; in a world of violence, loveless marriages, and women used for alliances, the short-lived marriage of Sansa Stark and Ramsey Bolton was still in a league of its own. There was no actual care here, as Sansa was persuaded into the union by the devious Littlefinger while on the run for her life, and Ramsey was using Sansa for her name and ability to give him a son. Ramsey was also one of the most brutal characters on the show - he humiliated her, abused her, raped her, and locked her up. There's little doubt he deserved his final fate, but no one can claim that a successful marriage ends in the wife feeding her abusive husband to his hounds.
Sansa and Joffrey
Sansa, heartbreakingly, managed to have two of the worst relationships in the entire series - no wonder she chose to rule the North as a solo Queen when she could. In the earliest seasons of the show, Sansa was betrothed to Joffrey Baratheon, and at first it was everything she wanted. She would marry a young Prince and become Queen - but that didn't happen. Joffrey was vicious and sadistic, and humiliated and tortured her before casting her aside for another woman and marrying her off to Tyrion.
Cersei & Robert
Cersei Lannister and Robert Baratheon didn't have the worst marriage, on the surface. In the first episode, it appears as though they are simply a mis-matched pair, but they are married, have been for a long time, rule together, and have many children.
Of course, the longevity turns out to be about the only part of that that is actually accurate, as it is revealed the Cersei is in love with her own brother, the 'Baratheon' children are actually his, and they have vicious fights that occasionally end in violence. And of course, Cersei has him killed, to set herself free.
Jon & Ygritte
Jon Snow and Ygritte do better than most couples on the show, as they are actually in love with one another - which is surprisingly rare. This sweet pair was a fan-favorite, as Ygritte showed Jon how to be a Wildling, and they became inseparable... until he betrayed her for the Night's Watch. Their success as a couple was shot down, first by her arrows into him when he left, and then by the arrow that killed Ygritte at Castle Black. This could have been one of the best, if Jon had only chosen differently.
Jaime & Cersei
It may seem strange to consider a damaging, incestuous relationship successful on any level, but this is Game of Thrones. And these two did 'love' each other, and as much as their relationship failed over the events of the series (as Jaime realized how vicious Cersei really was), they had a seemingly consistent love for decades before that, along with multiple children. They even chose each other at the end, and died in each other's arms. Disturbing as the entire situation is, had they not been related, this would easily have been one of the better relationships in the show.
Jon & Daenerys
Possibly the biggest relationship in the series was that of Jon and Daenerys Targaryen, even though this was another example of incest (although thankfully, this was an unknowing one, at least at first).
And for a while, these two were set to be an incredible power couple for Westeros - both driven, capable, strong leaders, and actually in love with each other. Of course, in Westeros, there seem to be no happy endings, as it was revealed that Daenerys was his aunt, and then that she went mad with power, and he ended up killing her.
Robb & Talisa
Talisa met Robb Stark on the battlefield, when he was already pledged to marry another - not the most auspicious beginning for their relationship. However, these two were actually quite successful, by Westeros standards. They fell in love, married, Talisa became pregnant, and they seemed set to become a powerful and capable King and Queen. They would rank higher, had they survived longer - but both were killed at the Red Wedding. At least neither of these killed the other!
Ned and Catelyn
Although their marriage was far from perfect, Ned and Catelyn Stark were a happy and successful pair... right up until Ned was asked to become King's Hand. The two married seemingly happily, and worked well together as the de facto rulers of the North. They had multiple children, were decent parents, and lived a good life. The only real issue here was that of Jon Snow, and his hidden parentage, which was a sore spot for Catelyn.
Arya and Gendry
It may seem strange that a relationship that only became romantic for a single night would rank so highly, but by all measures, they are actually extremely successful. They were allies and friends for years before things got physical, they both cared for each other, they both got what they wanted from the relationship (although Gendry arguably wanted more - or thought he did), and they both outlived the relationship (and the series). By Westeros standards, that's pretty impressive!
Most: Gilly And Sam
Their relationship may not be as tumultuous as many others in the series, but there is little doubt that it is one of the most successful. The moment Sam met Gilly, he fell for her, and saved her from her life with her father. From there, things became complicated - he struggled to find a way to keep her at Castle Black, to introduce her to his parents, and to find a way to tell her how he felt. By the end, though, this became one of the happiest endings in Game of Thrones. The two are together, raising Gilly's son and soon to raise their own, having survived everything Westeros had to throw at them and remained in love. Awwwww!