I'm inclined to think of it through the characters involved. Martin's surprised me before, but still:
All the "adult" living Stark children except one have become magical characters, and the fourth character is currently being groomed by the smartest villain in the seventh kingdoms, which seems likely to end with her learning all his tricks and then taking revenge on him.
In other words, while the series is known for the killing of the pure hero in Ned Stark, which would seem a "gritty" amoral emphasis, karma seems to be helping Ned's kids in epic quantities. When you also are aware that a tentative title for an upcoming book was A Time for Wolves, I would bet that by the end of the series, each of the Stark children is on the cusp of achieving "Legend of the Ages" status. What will that look like?
Jon appears to be the Messiah. I would expect him to become the primary force which beats back the White Walkers. It's possible he dies permanently at the climax, but only then.
Bran is the Greenseer. He will not be the primary force in the story that gets told, but it will be clear that he already has great power, and that that power will grow far stronger over time. He will arguably be the character of this era who has the most impact on the world in his lifetime.
Arya will be soon be a full-on magic assassin AND re-united with her direwolf who has amassed a wolf army. I'd expect her to come back to Westeros, play a key role in helping the good guys win the big battle, and then go off on her own again (with or without her wolf). She'll be talked about as a scary legend that parents tell their kids so they don't go off into the woods.
Sansa becomes a super politico, takes down Littlefinger, marries a blueblood of decent character, but continues to be the one who wears the pants (marrying Tyrion seems a bit too tidy).
Figure, Catelyn the undead mother sticks around long enough to see the sun shine again ... with bloody results.
Where does that leave everyone else:
Well, I'm expecting Dany to either sit on the Iron Throne, or to be married to Jon and have him sit on the throne. Gun to my head, I'm thinking she's on it and Jon's dead, but I do expect the two of them to come together at a certain point, and quite possibly breed.
Cersei is dead and defeated.
Cersei's adorable kids, and the Tyrells, I think they'll be dead and defeated too, and these will be the saddest deaths around. They just have too much to lose in accepting a demotion for it to come all that rationally.
Jaime, well, I think Catelyn kills him. His transition toward being a decent person appears to be moving him toward being truly heroic, but I don't think he gets that far. Trying to kill a baby Stark is a bit too much to come back from.
Tyrion is Dany's Hand through the conflict, and either continues on with her, or goes home to rule the Lannister clan from their home base.
Varys remains a power player through his connection to Tyrion, and indirect connection to Dany.
Stannis is dead and defeated, though not everything associated with him will be. Those he has sent away, including Davos, will have a future. House Baratheon may live on.
Melisandre plays a crucial role with Jon and then either dies, or goes off on her own again, possibly back to Essos.
The Martells are thriving, having come back to the side of the dragons early on.
The current Greyjoy leadership is humiliated, I'm undecided about what that will mean for Asha. It's possible she inherits the Iron Islands by being a convenient ally at an opportune time. Theon seems likely to come full one eighty and die heroically, possibly playing a crucial role, saving his sister.
Rickon Stark becomes the new lord of Winterfell. Seriously, I can't imagine why this character even exists except to have a remaining Stark still around to keep Winterfell in the family. We haven't dealt with his shattered, violent psyche though. The future of Winterfell could prove to be a gradual descent away from what Ned stood for.
The Citadel falls, and we see Samwell and company do various scrappy things to survive. The Martell girl in disguise becomes a fan favorite.
Lastly, what about Aegon? I just have a hard time believe that he'll accept a place secondary to Dany or anyone else, and an even harder time imagining an ending where he ends up the big "winner" of all this conflict. I expect him to end up either dead or in effect exiled back to Essos Blackfyre-style. Very solid chance he ends up a fake who has none of the dragon powers Dany does.
You'll notice I said nothing about who rides Dany's other two dragons, which a lot of people are focused on. To be honest, this whole interspecies 'shipping' makes me cringe a bit, partly because it seems trite, and partly because I am not encouraged about how it will play out.
The three biggest characters in the story appear to be Dany, Jon, and Tyrion, with Tyrion being by far the most major by page-time and by far the most beloved, but also by far the least important to the future of Westeros.
My sincere hope is that Martin resists the temptation to make Tyrion into something more glamorous than he already is ... which he might do by making him a dragon rider. I'd have more faith in Martin showing such restraint though if I could think of other candidates to ride the dragons. I really have no idea, no major character put into those roles seems appropriate to me.
So, I'll draw the line above. Despite the huge amounts of violence involved in what would require an ending like this, it does feel to a certain degree too tidy. I kind of expect that Martin sees this as the tidy solution, and that he'll keep this at least as a possibility until the very end, at which point he might twist abruptly away from it.
Foremost in the temptation here: The solution I've describe above implies that things largely go back to just the way they were before. I imagine Martin would like to leave the land permanently changed. Martin could do this in a variety of ways, but the simplest way to pull it off while keeping much of the fan-invested characters in deep catharsis would be:
Destruction of the wall, and a re-merge of the seven King & Wildling cultures. Mance Rayder is a bit too awesome of a character to go away, so he may end up on top in the north ruling in a way that is still far more loose than what gets done down south.
Mance & co will have strong relationships with the Children of the Forest through Bran, and a new security against the White Walkers will be achieved through this, rather than the apartheid that existed before.
More questions on A Song of Ice and Fire (book series):