EPTA bit on my background regarding the series. I'd guess I'm the fairly "average, invested" fan. I came to the series late, having started it about a year ago now. The advantage this gave me was that I could read the entire (current) series all the way through, without stopping between books. I read the first 4 in about a series of 3ish months, took about a month break, then finished off a Dance With Dragons about a month after. I have a pretty good memory, and reading the entire series through was helpful in picking up on a lot of the minor characters and keeping the events pretty straight in my head, in terms of when they occurred. Additionally, I've spent probably too much time reading the ASoIaF wiki, so I'm pretty current on a lot of the theories etc. swirling around the series (the wiki is also great for details and brushing up on past events, and events that happen outside that series). I've also been watching the HBO tv show, and am almost caught up on that (I'm on the 2nd episode of season 3). I've been watching it for an eye as to what they change regarding the books, specifically keeping in mind that they hopefully wouldn't remove anything that would be critical to include for future plot twists. That said, most of what I'm going to say isn't really novel or break-through; more it is a synthesis of a lot of the good ideas floating around out there (on teh interwebz) with my own thoughts sprinkled in. So here is what I'm currently thinking is in the future for this series:
1. At this point, I'm pretty sure Jon Snow is the son of Lyanna and Rhaegar. I don't really like this, and kind of hope that it won't be true, but both the evidence in the book and "reading between the lines" in the show indicate this is probably true. He is possibly even Azor Ahai and The prince that was promised. He is the son of ice (stark) and fire (targaryen), and this song is primarily about him. He may or may not still be a "bastard" depending on whether Rhaegar actually married Lyanna. I think if there is a "hero" that will fight against the Others, he already needs to be well established in the series for the audience to have an attachment to him. The list of possible candidates is pretty slim, with Stannis being unlikely given that his Lightbringer doesn't appear to be the true lightbringer (doesn't give off heat). If Jon does turn out to be this person, look for craziness including "warging" into dragons, and other magic shenanigens. He may end up getting his hands on Stannis's Lightbringer and killing Daenerys with it to make it the "true" lightbringer (if this happens, it will be very near the end of the story).
2. Jaime will be responsible for the death of Cersei, possibly and hopefully, will directly kill her. This is becoming fairly obvious by the end of DwD, so this shouldn't be that surprising. Cersei is haunted by the prophecy that she received when she was a kid, of being ruined by a "younger, more beautiful queen (probably Daenerys)" and being killed by the valonqar. She thinks Tyrion is the younger brother, but it is actually Jaime, who is also her younger brother, technically. Additionally, expect either Brienne to die for Jaime, or Jaime to die for Brienne (or possibly both).
3. This isn't as much as a prediction, but more of a concern. The series contains 2 major story arcs: the fight for the Iron Throne of Westeros, and the battle against the Others. Sometimes GRRM tries to combine the two, and I think he is still trying to, but honestly the battle against the Others feels like it has been given short shrift. I'm starting to get concerned that he either won't give it the proper amount of development, or just won't finish it at all (either by decision or not being able to finish it), and leaving it for a possible future author to handle.
There are still a lot of loose ends and questions, though. Obviously, Arya is going to continue to go on some form of killing spree, but to what end in the long term I don't know. I'm pretty sure at this point Littlefinger just wants to destroy all the major houses of Westeros, especially the Starks, Tullies and Lannisters (this is more obvious in the tv show than the book). I doubt he will succeed, but he'll probably do a lot of damage before he is stopped. How the Iron Islands and that whole story fits in, I'm not sure. What is going in Oldtown with the Faceless One, and who he has to kill (and he might even be after some item of importance) I don't know. I imagine Tyrion will end up joining up with Daenerys, but to what end I don't really know. I feel like a lot of the new characters (sand snakes, the young Aegon, Jon Connington) are just distractions to be killed off by more major players, but I'm sure there is a twist or two in there among them.
On a side note, I find it pretty amusing that the last time we saw Jon Snow, he seemed to be dying, but basically no one I've talked to or read seems to think he is actually dead. There are some interesting theories regarding how he might survive, ranging from him "warging" into Ghost or maybe the giant, or just being saved by other Night's Watch brothers.
