Book and Show Spoilers follow:
Jon Snow's Parentage
Introduced In: Season 1/Season 6
Estimated Resolution: End of Season 6
It's almost a foregone conclusion that Jon Snow is not actually the bastard son of Ned Stark, but in fact the secret son of Rhaegar Targaryen and Ned's sister Lyanna. At this point, it would almost be a disappointment for this to resolve in any other way ... and thus it would be best for the show to get this confirmation out of the way, the quicker the better.
House Lannister v. House Stark
Introduced In: Season 1
House Lannister v. House Stark has been an ongoing plotline that has occupied a large portion of many seasons of the show (and the novels). It was apparent from episode 1 that these Houses would not get along (given the closing of the first episode features Jaime Lannister throwing Bran Stark out a window for discovering his and his sister's secret twincest shame). This animosity quickly descended into Ned Stark's beheading by Joffrey Lannister and the war of the Five Kings that strongly featured major conflict between a Robb Stark and Lannister forces. This conflict culminated in the Red Wedding, where Lannister surrogate Walder Frey viciously murdered Robb, his new wife Talisa, and Catelyn, matriarch of the Starks.
Estimated Resolution: Season 8
Yeah, so the Lannisters have pretty much been winning this House rivalry contest for a while now. That said, how does this plotline NOT go down the path of Stark revenge against the Lannisters? I mean, I know the Lannisters have taken a few hits (Seems like all the Lannister Baby Boomers are gone), but their more prominent members still hold an iron (golden?) grip over the Iron Throne (Cersei, Jaime, and to a lesser extent Tommen), even if that grip is tenuously shared with the Tyrells - and now the Faith Militant. And with Sansa Stark and Jon Snow maneuvering themselves into alliances in the North, and Arya on the cusp of completing her assassin training, it seems like some good ol' comeuppance is about to be served.
As an antagonistic dynamic that has spanned the show's entire existence, the culmination has to happen in Season 8 ... right? I'm guessing the Starks and Lannisters have to set aside their differences to fight the White Walkers ... until one betrays the other and starts another murder spree.
Speaking of Arya ...
Arya and the House of Black and White
Introduced In: Season 2
While masquerading as a servant in Harrenhal, Arya meets Jaqen H'ghar, who first displays the power of the House of Black and White to her. Upon escaping the Hound (and Brienne), Arya arrives in Braavos in season 5 and began a multi-season foray into becoming an assassin and devotee of the Many-Faced God (aka the God of Death). Her efforts have been plagued by moral dilemmas, the forced abandonment of her identity as a Stark noble, punitive blindness for indulging her need for revenge and the Waif, aka that high school bully that likes to pick on you just because you exist. Recent events have led Arya to stray from her assassin training and her mentor Jaqen H'ghar, and her latest step - the retrieval of her rapier Needle, the strongest physical tie she still maintains to her true family - only serves to highlight the fact that Arya was never going to become a true assassin.
Estimated Resolution: Season 7 / 8
And in all honesty, who really wanted to see Arya become a fullblown assassin for hire? What we really wanted was for her to become an epic ninja and use her new skills to cross everyone off her list. Season 6 has been telegraphing Arya attaining that expertise and then abandoning the House of Black and White for a while and seems on track for an epic showdown between Arya, the Waif, and Jaqen for the current season finale.
While I've no doubt that Arya will escape the grasp of the Many-Faced Gods and his assassins in the current season, it seems suspect that she could truly free herself from such a powerful and dangerous organization without suffering some kind of retribution for her repeated failures. My bet's on a second confrontation with the House of Black and White in season 7 or season 8, where we can expect Arya's visage to join the Hall of Faces.
Introduced In: Season 5
Cersei has been portrayed as a paranoid, vindictive, ruthless, and power-hungry Queen. It's arguable that these personality traits were inceptioned into her - or at the very least exacerbated by - the prophecy she heard as a young girl. Maggy the Frog predicted that she would marry the King - not a prince as she expected. That she would be Queen, until a younger more beautiful Queen came to take her place. That she would have three children, but that they would all die before her. There are subtleties and additional predictions made in the novels, but it's these predictions that have shaped show Cersei the most.
Estimated Resolution: Beginning of Season 7
Parts of these prophecies can be said to have already taken place. For example, we know that Cersei did in fact marry King Robert Baratheon after he usurped the position from the Mad King Aerys Targaryen. We also have already seen her lose her title of Queen to Margaery - though I suspect Danaerys is also a contender for this position depending on Margaery's fate. Finally, two of her children have already been slain - Joffrey by the schemes of Littlefinger and Olenna Tyrell, and Myrcella by Ellaria Sand to avenge Oberyn Martell. Only Tommen remains alive ... and with the High Sparrow, the Tyrells, and his Lannister parents all vying to control him, how long can we expect him to live? I say early season 7 at the latest.
Martell v. Lannister
Introduced In: Season 4
The Dorne plot came into prominence in Season 4 with the arrival of Oberyn Martell and his lover Ellaria Sand in King's Landing for Joffrey's wedding. Oberyn subsequently gets himself involved in Tyrion's defense when our favorite dwarf is accused of orchestrating Joffrey's assassination. Of course, Oberyn Martell looks poised to deliver a flawless victory until his own arrogance and need to pontificate over the Mountain's beaten body gets himself famously eye-gouged and head-crushed. Leave it to the Mountain to go out with a bang ... or disgusting squelchy noises, anyway. Since then, Ellaria has been rousing allies in Dorne to avenge Oberyn's death.
Estimated Resolution: Not Soon Enough, but let's hope for the end of Season 6
BLURGH. Can we be done with Ellaria and her sand snakes already? We get it, Oberyn was awesome ... but he also kinda asked for it. He volunteered to defend Tyrion of his own volition. He lost because he was overconfident. While the Dorne plot in the books is layered with more subtleties (and less leaps of logic like how the lover of one of Dorne's princes could somehow rouse enough support to seize command of an entire kingdom), neither the book plot or the show plot for this Dornish invasion has been very compelling. On the show especially, it's hard to care for Ellaria or her Sand snakes. Maybe Ellaria's army and the Faith Militant can murder each other, that would be nice . Two birds with one stone! Let's put this ish to rest already.
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There are of course tons more plotlines that have yet to be tied up, but that's it for today! What do you think? Did I get any details wrong? What plotlines do you see coming to an end soon?