My theory on the prophecy "The dragon has three heads"

YES!  I believe that Daenerys, Jon Snow and Brienne of Tarth are the 3 dragons! See why..

Clue 1 Family Tree. 

They all share Targaryen blood and they are the only ones left alive, apart from Rhaegar's son if he is Aegon) As you will read in previous posts I explain why I believe Brienne of Tarth and Jon Snow are related to Daenerys.

Clue 2 Martin's word

The author suggests that not all heads are Targaryens. Apart from Daenerys the other two does not share the last name.

Clue 3 Personality

All 3 characters are heroes, they share a sense of justice and morals. They truly want to serve humanity and the 3 heads need to have a common cause to fight for in order to be in balance.

Clue 4 Prophesy rejects all the rest

"Soon come the pale mare, and after her the others. Kraken and dark flame, lion and griffin, the sun's son and the mummer's dragon" Quaithe prophesy to Daenerys.

This prophesy rules out both Tyrion aka "lion" and Aegon aka "mummer's dragon"

Brienne of Tarth: Grand-daughter of Duncan the tall and Daella Targaryen?

Brienne of Tarth: Grand-daughter of Duncan the tall and Daella Targaryen?

First Clue: Appearance/ Personality
Dunk, later in life known as Ser Duncan the Tall, was a large man who grew up and became the most trusted companion of Aegon V Targaryen (aka "Egg"), and served as Lord Commander of the Kingsguard during the his reign. Nearly seven feet tall, Dunk had a humble personality and thought of himself as thick-headed. His early exploits are chronicled in the "Dunk and Egg" stories of The Hedge Knight, The Sworn Sword, and The Mystery Knight. His personal arms were a green shooting star above an elm tree proper on sunset.  
Second Clue: A family heirloom
A shield with Dunk's original arms came to reside in the armory of Lord Selwyn Tarth. Selwyn's daughter Brienne wore a copied version of the arms while traveling the Riverlands incognito.
Third Clue: Family Tree
According to the books, "Tarth boasts of blood-ties to the Durrandons, Baratheons and more recently the Targaryens", which suggests a marriage between a Tarth member (Selwyn Tarth) and a Targaryen descendant (Bryenne's mother). The only Targaryens that have unknown descendants are Egg's sisters, Daella and Rhae so it is possible that Dunk fell in love and married one of Eggs sisters and that child married a Tarth.
Fourth Clue: Martin's word
The author confirms that a character from the book is a descendant of Duncan

Daenerys Targaryen is the aunt of Jon Snow theory

During the tourney at Harrenhal that was hosted by Lord Walter Whent to celebrate the name day of his maiden daughter, the Starks met  Howland Reed  who was bullied by rowdy squires. Lyanna Stark annoyed by their behavior tried to protect him and  later disguised as  a mystery challenger, known as the Knight of the Laughing Tree entered the joust to defend the honor  of Howland Reed by challenging and defeating the three knights whose squires were the once that bullied him. After that the knight disappeared  and an angry king Aerys, who was certain that the man was his enemy, sent Rhaegar to search for the vanished knight, but Rhaegar reported that only his shield could be found, abandoned in a tree. 

My theory is that though the knight's identity remains unknown, Rhaegar actually found that Lyanna was the Knight of the Laughing Tree but he didn't wish to give her identity away, from fear of his father. Still he was so impressed by her that after winning the tournament, Rhaegar gave her the winter rose instead of his wife, not to crown her queen of love and beauty but out of respect because he felt that she should win the tournament. This gesture however was misinterpreted by most of the crowd, as Lyanna was engaged to Robert Baratheon and Rhaegar was married to Ellia Martell. 

After that Lyanna and Rhaegar gradually fell in love. Lyanna, sharing the personality of Aria, was never happy by her engagement with Robert Baratheon, as she felt that Robert was only attracted to her beauty while Rhaegar loved her for her boldness, her rebel spirit and all those things a Lady is forbidden to be.  My theory is that they shared a night together after the tournament. It is difficult to imagine them having secretly formed a relationship, if you think the miles that separated them. 

However after that night Lyanna was pregnant and desperate to protect her child from her fiancee she sent a raven to Rhaegar. Rhaegar found it out and in order to protect her and their child he convinced her to run away with him before her belly showed any signs of pregnancy. Lyanna's family not knowing anything though thought that Rhaegar kidnapped Lyanna, for reasons unknown and this action led her angry fiancee Robert  to start a war, accompanied by the Starks. 

Howland Reed fought alongside Eddard Stark during Robert's Rebellion, and he accompanied Eddard to the Tower of Joy  to reclaim Eddard's abducted sister Lyanna, who mysteriously died at that time covered in blood.My theory is that Lyanna died while giving birth to John Snow, the son of Rhaegar and made Eddard to promise her not to uncover his identity to anyone because Robert would kill the child. Howland is apparently the only remaining person to have witnessed the mysterious events that took place in the tower that day and therefore the only person that knows who were the true parents of Jon Snow.