According to Reddit user Zahn1138, the rightful heir to the Iron Throne isn't Daenerys Targaryen, Jon Snow, Tommen Baratheon or Gendry.
The rightful heir, by way of a very detailed explanation, to the Iron Throne and the ruler, by right, of the Seven Kingdoms is none other than Brienne of Tarth.
I can hear your gasps through the screen and see you shaking your heads, but just hear me, or should I say Zahn1138, out.
According to Zahn1138, here's what we all currently know regarding the rightful claim to the Iron Throne:
1. The House Targaryen are the rightful Kings of the Andals and the First Men, and Lords of the Seven Kingdoms. 2. The royal dynasty of House Baratheon, comprised of Kings Robert I, Joffrey I (false-born), Tommen I (false-born), Renly I, and Stannis I all based their claims on descent from Princess Rhaelle Targaryen, daughter of King Aegon V Targaryen, and direct patrilineal descent from Aerion Targaryen, father of King Aegon I Targaryen, through Aerion's bastard son, Orys Baratheon. 3. The House Baratheon is now all but extinct, with Gendry the Oarsman as the only known survivor (in the show). He and his half-siblings from the books are bastards who have not been legitimized, so they have no claim.
Now, here's where things get a little interesting, if you follow this user's explanation.
4. The House Tarth has recent descent from a Targaryen Royal - 'Tarths boast of blood-ties to the Durradons, Baratheons and more recently the Targaryens.' 5. The House Tarth proudly displays the arms of their forebear, the noble Kingsguard Ser Duncan the Tall, confirmed as an ancestor of Brienne Tarth, whence come her immense size and skill in battle. 6. The daughters of King Maekar I Targaryen, the princesses Rhae and Daella), are both confirmed to have wed and had children, testified by their brother Maester Aemon).
OK, so now we all can see it's plausible in more ways than one that Brienne of Tarth is indeed a direct descendent of Kingsguard Ser Duncan the Tall.
Now, here is the legitimacy of the Tarth line, including Lady Brienne of Tarth:
7. Ser Duncan was born in 192 AC, Lord Selwyn Tarth in 245 AC, and Brienne in 280 AC. This timeline makes it most likely that Ser Duncan is Brienne's great-grandfather, and Lord Selwyn's grandfather. 8. House Tarth would not display Ser Duncan's arms with pride if their descent from him was from bastardy or cuckoldry. It must have been legitimate descent. 9. House Tarth claims unbroken patrilineal descent from their founder - a recent marriage to House Targaryen must have been of a Lord Tarth to a Targaryen princess. 10. The only unaccounted Targaryen descendants since King Daeron II, are through Aegon V's daughters Rhae and Daella, both confirmed to have married and had descendants. 11. Since both Rhae and Daelle would be in their forties for Lord Selwyn's birth, Lord Selwyn is likely one princess's grandson. 12. Since Lord Selwyn possesses and displays Ser Duncan's arms, he is Ser Duncan's heir. 13. As Lord Selwyn is Ser Duncan's heir, it is through his mother, as House Tarth would not proclaim its cuckoldry were Ser Duncan his natural father, and Ser Duncan is too young to be his great-grandfather. 14. Lord Selwyn's maternal grandfather is thus Ser Duncan, as his paternal grandfather was a previous Lord Tarth.
Are you still with me? Here's where it appears to become a little clearer as to how the Tarth line intertwines with King Aegon.
15. Ser Duncan was intimate friends with Aegon V, and knew his sisters well. 16. Daeron II's children were known to marry for love, such as Aegon V's wife, Betha Blackwood. 17. The simplest explanation is that Ser Duncan wed either Rhae or Daella, and that his daughter by one of them was wed to Lord Tarth, becoming Lord Selwyn's mother, and Brienne's grandmother.
Now, here appears to be reasoning for why Dany and Jon Snow don't have any legitimate claims to the Iron Throne:
18. The claim of Prince Maegor), firstborn son of the deceased Crown Prince Aerion, was denied by a Great Council despite his right as firstborn son of the firstborn son of the King - why? His madness. 'fears that the boy might have inherited his father's cruelty and madness.' (TWOIAF, p. 181) 19. King Aerys II, known as the 'Mad King', unseated the Targaryen Dynasty as a result of his madness. 20. By the Great Council's precedent of denying primogeniture on the basis of inherited insanity, all of King Aerys II's descendants are ineligible to rule, disqualifying claims from Jon Snow, [Daenerys] Stormborn, and Aegon VI (book only).
Finally, here's why Brienne of Tarth does have a legitimate claim:
21. The only known noble who is of legitimate Targaryen descent, and whose claim is not nullified by descent from madness is Lord Selwyn. 22. Lord Selwyn is the rightful possessor of the Iron Throne, King of the Andals and the First Men, Lord of the Seven Kingdoms, and Protector of the Realm. 23. Brienne is his only heir, and thus the rightful heir to the Iron Throne.
This is wild, but if you take the time to read and research everything outlined in the above theory, it actually makes perfect sense. Brienne of Tarth is the rightful heir to the Iron Throne.
Reddit user Zahn1138 concludes,
Remember, the prophecy is of 'The Prince(ss) that was Promised', not 'The King/Queen that was Promised.' If Jon's claim is true, he is the King. If [Daenerys'] claim is true, she is the Queen. Only Brienne is yet a Princess, as her father yet lives. She received Oathkeeper from a man she loved (Ser Jaime Lannister), who she is now on the way to fight. She will plunge it into his heart and draw it out as Lightbringer. Brienne = Azor Ahai confirmed.