Aegon was the second child of Rhaegar Targaryen and Elia Martell; just an infant in the Sack of King’s Landing. We learn from numerous sources that Gregor slammed Aegon’s head against the wall during the sack, crushing his skull and killing him. Gregor himself confirms this in his duel with Oberyn Martell.
However, it has been suggested that Aegon is still alive, possibly smuggled out of the city somehow, and the baby killed by Gregor was a substitute. Supporters of this theory think the grown Aegon will reappear and possibly become one of Dany’s heads of the dragon.
The problem with this theory is that there is zero evidence to support it. However, GRRM himself has written several tantalizing letters on the subject. For example, take SSM # 100
Q:Are Aegon and Rhaenys, Elia’s children, well and truly dead?
A: All I have to say is that there is absolutely no doubt that little Princess Rhaenys was dragged from beneath her father’s bed and slain.
Note that GRRM does not confirm Aegon’s death. However, he does not deny it happened. From his wording, he is merely saying that there is doubt that Aegon was killed. This could mean everything or nothing.
However, there might be a chance that Aegon is still alive. Could we have met him already? We know Aegon looked like a Targaryen, with silver hair and violet eyes.(SSM, #114) We also know that, because he was an infant during Robert’s Rebellion, would be about thirteen or fourteen at the beginning of Game of Thrones.
Who fits this description? One suggestion has been Samwell Tarly, evidence being that his bookishness and intelligence might have been inheirited from Rhaegar. Also, Sam is described as having ‘pale eyes,’ which may or may not be pale violet like other Targaryens.
However, Sam does not fit the description. There is no mention of Sam having silver hair, which is such a rare trait that it surely would have been brought up at some point. Also, Sam is a little too old (16 years old in CoK) to be Aegon. Finally, there is no plausible reason why whoever smuggled Aegon out of the city would place the baby in the care of the gruff Randyll Tarly, and also get Lady Tarly in on the secret.
Another possibility that comes up from time to time is Edric Dayne, Beric's young squire who Arya meets in SoS. Edric suspiciously has pale silver hair and "big blue eyes, so dark they looked purple." (SoS, pg. 493). These sound like Targaryen traits. Could Edric be the long lost Aegon?
There still remains one problem. Edric is only 12 years old. Since Aegon was an infant during the Sack of King's Landing, he must be at least 15 or 16 in the present day. There is a big different between a 12 year old and a 16 year old. Thus, Edric appears to be much too young to qualify as Aegon.
Other suggestions have been varied and implausible. One of the many suggestions that sprouts up from time to time is Daario Naharis, the sellsword who joins up with Dany in SoS. However, since Daario would have to be 15 or 16 at the oldest to be Aegon, this does not seem likely.
So if baby Aegon is still alive and out there, he is most likely a character we have yet to meet. While the ‘Aegon is Alive’ theory has merit and nothing going directly against it, there has not been any evidence thus far to support it, as opposed to most of the other theories presented at this site. So, as of now, the question is still very well debatable and up in the air.