|Theories may be removed if ...|
A source of speculation that has existed for years within the fandom is whether or not Aegon Targaryen survived. A Dance With Dragons reveals that the character is alive and living under the false name of Young Griff in order to hide his true identity. A number of fans believe the character to be a fraud.
Did Aegon survive?
Fans have wondered for years whether or not Aegon survived which was spurred on by George Martin not confirming that the character died in one of his communications. He did conform that there was no doubt that Aegon's sister died. 
Edric fits the description of what Aegon would look like with his purple eyes and pale blond hair. Yet, he is 12 years old in A Storm of Swords, which is too young.
The evidence for Sam includes being that his bookishness and intelligence which might have been inherited from Rhaegar Targaryen. Also, Sam is described as having ‘pale eyes,’ which may or may not be pale violet like other Targaryens. However, there is no mention of Sam having silver-gold hair, which is such a rare trait that it would have been noted.
Aegon Targaryen/Young Griff
In A Dance With Dragons, Tyrion Lannister discovers that a youth going by the name of Young Griff is truly Aegon Targaryen in disguise. A story is given of how Aegon was switched with another baby and spirited away to be raised in secret until he could claim the Iron Throne. He fits the description of Aegon with purple eyes and silver blond hair that he dyes blue.
Not all fans believe he is the true Aegon however.
Theory of why Young Griff is fake
Reasoning against this theory would be Varys' speech where he stated to the dying Kevan Lannister:
"Aegon has been shaped for rule before he could walk. He has been trained in arms, as befits a knight to be, but that was not the end of his education. He reads and writes, he speaks several tongues, he has studied history and law and poetry. A septa has instructed him in the mysteries of the Faith since he was old enough to understand them. He has lived with fisherfolk, worked with his hands, swum in rivers and mended nets and learned to wash his own clothes at need. He can fish and cook and bind up a wound, he knows what it is like to be hungry, to be hunted, to be afraid. Tommen has been taught that kingship is his right. Aegon knows kingship is his duty, that a king must put his people first, and live and rule for them". 
Since Kevan was dying, some speculate that there was no reason for Varys to lie. But then, this speech does not imply Aegon is a Targaryen at all, but instead that he is simply well qualified for the throne, as Tyrion remarks to himself " the perfect prince".
Quaithe telling Daenerys to beware "the mummer's dragon". Either this refers to Aegon being under the control of "the mummer" - Varys was a mummer in his childhood and is a master actor - or that he's a false dragon, a fake Targaryen.
A key problem with the pisswater prince story is that it should not have been possible for Varys to know in advance that Aegon would be killed in such a manner as to leave him unrecognizable. If the pisswater prince's face had remained intact after his death as Rhaenys' was, no one would have been fooled. Moreover, the Valyrian features of silver-gold hair and purple eyes are so rare in Westeros that it is extremely unlikely that the pisswater prince could have had them. Pale blond hair may pass for silver-gold if there is very little left and it's covered in blood, but Varys could not have known in advance that Aegon's whole head would be smashed. The pisswater prince plot only makes sense retrospectively, knowing that the baby presented by Tywin was not recognizable as Aegon. It seems more likely that Varys cooked up this plot after the real Aegon had died to exploit the shred of doubt that the corpse's condition had created.
Possible True Identities of Young Griff
- A Blackfyre: Illyrio tells us that the Blackfyres are extinct in the male line, implying that some female-line descendants may still be extant. If Aegon is actually a Blackfyre, this would explain the Golden Company's allegiance to him.
- Son of Illyrio Mopatis: Illyrio is noted to have had a wife with Valyrian features in the past, it may be that Aegon is her son by Illyrio. Illyrio may have even married a Blackfyre descendant who would have been a granddaughter or great-granddaughter of Daemon (whose heirs besides his first few children and his grandson Maelys are unknown). This would explain why Illyrio was seemingly eager to help the Viserys and Danaerys to make their claim for the Iron Throne, yet apparently wanted nothing in return. He and Varys may have had no real intention of crowning either, but they needed Viserys's claim to build an army in advance of crowning Aegon. They may have intended to kill Viserys and Danaerys, or they may have expected them to be killed by one of Viserys's many enemies.
- A random boy of Valyrian descent: The Valyrian features of silver-gold hair, purple eyes and unearthly beauty are sufficiently common in Essos, especially in Lys, that it should not have been hard for Varys and Illyrio to find a suitable baby to raise as a fake Aegon to claim the Iron Throne.