WARNING: SPOILERS AND MAYBE SOME HALF-ASSED PREDICTIONS AHEAD!
1. JON SNOW
113 total scenes
Most of the promotional material I've seen for GoT S8 has featured the lovely mug of Kit Harrington's lovable bastard... OR IS HE? I mean, he is lovable either way, but his lineage has been thrown into doubt with some conveniently-placed flashbacks. He's taken over the Night's Watch, recruited the Free Folk, and made an alliance with the Dragon Queen (i.e. the queen who has dragons, not the one who acts like a dragon). His upward trajectory and unwavering good alignment make Jon Snow a powerful protagonist, but will he feel any ill effects from the dark magic resurrection spell from the end of Season 5?
2. TYRION LANNISTER
109 total scenes
I have been known to clown Peter Dinklage's accent in the show, since as a straight-up British accent, it honestly isn't all that sharp. But that was in my younger, more judgmental days, and now I realize that his character's not British, he's not British, and not even all the people who play Lannisters on the show are British. It's an accent that is sui generis, and either way, it hasn't distracted me from enjoying the quality of Dinklage's acting through the years. Also THE DINKLES THO!!!1
As one of the prime behind-the-scenes movers in GoT, Tyrion has never exhibited the traits of a true power seeker, and he seems more like a support class than a classic hero.
3. DAENERYS TARGARYEN
101 total scenes
Khaleesi was a fan favorite from the moment audiences saw one of Emilia Clarke's 12,000 wigs (am I getting that right?). Westeros history buffs are also more likely to root for her, pointing out that her family was initially in charge of the Seven Kingdoms, until they were overthrown by the Starks and Baratheons. Although, who did the Targaryens take over the throne from when they emigrated from ... is it Easteros? Essos? The point is, you live by the coup, you die by the coup, and not all rebellions are bad as far as the PEOPLE are concerned. She's got dragons, an army, a love interest, a good advisor, and DRAGONS. Not to mention her goal is clearly the throne, so she's gotta be considered the number one contender.
4. ARYA STARK
76 total scenes / 75 primary scenes
5. SANSA STARK
76 total scenes / 62 primary scenes
It cannot be a coincidence that the two Stark sisters have appeared in the same number of total scenes over the course of seven seasons of television... OR CAN IT? Digging deeper into the numbers in the chart above, you can see that Maisie Williams's Arya has the advantage over Sophie Turner's Sansa in Primary Scenes (where a character's story was the primary purpose of the scene, from my unprofessional perspective), as opposed to Extra Scenes (where that character's story was advanced in a scene that primarily featured a different character). Either way, they are firmly back on the same side, manning the first line of defense against a zombie hoard, and starting further than anyone from the location of the actual Throne. Not looking terribly promising for the Winterfell crew.
6. CERSEI LANNISTER-BARATHEON
74 total scenes / 63 primary scenes
The only true villain so far on this list, Lena Headey's character is close in scene total to both Arya (in total scenes) and Sansa (in primary scenes), which makes sense, given all the scenes the two of them shared in Season 2. But the question is, should Cersei even still be referred to as Baratheon? Her Baratheon husband is dead, as are all three of the children that she was trying to pretend were Baratheon; might as well drop the charade, right? At any rate, in the eyes of succession nerds, her claim to the throne is at best suspect, but let's not forget that possession is 9/10ths and all that. Given that Cersei has all the resources of both King's Landing, the Tyrells, and the Greyjoy fleet, Daenerys would do well to remember that a rightful claim isn't power, POWER is power.
7. NED STARK / ROBB STARK
53 total scenes
8. STANNIS / DAVOS / MELISANDRE
50 total scenes
I'm grouping these two stories together because they're basically finished. Sean Bean, Richard Madden, and Stephen Dillane have been off the show for multiple seasons now. It's true that Carice van Houten's Melisandre has kind of become a free agent, and Liam Cunningham has gone on some solo missions since Davos Seaworth joined forces with Jon (most recently to pick up another throne claimant Gendry), but neither really drive the narrative at this point.
9. BRAN STARK
46 total/primary scenes
Since going north of The Wall, Isaac Hempstead Wright has gone full Druid, all but removing himself from the concerns of his now former family, the Starks. Bran now goes by the moniker "Three-Eyed Raven," he can see different areas of the timeline, and he has lost all sense of social interactions. Scrying, ahem, WARGING is a useful tool for any adventuring party to have, but it seems like Brandon has taken on more of a support role.
10. JAIME LANNISTER
46 total scenes / 45 primary scenes
This MUST be a coincidence, that the eldest surviving Stark son and the eldest surviving Lannister son appear in the same amount of scenes... OR IS IT!?!?!?!? We've seen a shift in Nikolaj Coster-Waldau's character from being purely power hungry and serving his own self-interests (thereby serving those of his family) to having some concern about the greater good. Will it be enough to aid in the battle against the darkness, or will the knight's change of alignment be too little, too late?
Above is the series cast summary for Game of Thrones on IMDB, as of just mere hours before the premiere, showing that the actors with top seven episodic appearances are also part of the top 10 in terms of story scenes. Of the remaining ones, Samwell Tarly (John Bradley, 38 scenes) is still in the mix, heading back to the action from Oldtown with some pertinent information. Theon Greyjoy (Alfie Allen, 37 scenes combined) has embarked on some sidequesting to rescue his sister Yara (nee Asha, played by Gemma Whelan, and who is also represented in the gray-shaded scenes). I had grouped Littlefinger (Aiden Gillen) and Varys (Conleth Hill) together, but they only truly carry 12 total scenes, since their two storylines were closely linked to Sansa and Tyrion, respectively. The same goes for Iain Glenn's Jorah Mormont, who had appeared in solely Daenerys-dominated scenes for the majority of his time on the show.
I know people always say that past results do not guarantee future performance, but you mostly hear that in terms of finance or athletic performance. Who knows if it applies to a narrative that's been designed for a particular resolution? Either way, now you know who's had the most, if not screen time, than STORY time in all the past Games of Throne, and it's up to you to decide how it might affect the unfolding season!