The most common suggestion for the third head is Tyrion.
scifi.stackexchange.com/users/23/Mike Scott
Now that A Dance with Dragons has been released
I believe it's worth adding another answer to this question.
A Dance with Dragons spoiler
It seems more likely now that one of the dragons is Aegon Targaryen.
Aegon Targaryen was the second child and only son of Prince Rhaegar Targaryen and Elia Martell.
It is revealed that the baby murdered during the Sack of King's Landing was that of a peasant who Lord Varys put in the cot.
Varys then hid baby Aegon.
When Robert Baratheon and the realm falsely believed him dead, Varys had agents smuggle him across the narrow sea.
Aegon_Targaryen @
asoif wiki .
Also It now seems possible, although not certain, that Jon Snow isn't one of the three; so the other answers could still be just as accurate.
scifi.stackexchange.com/users/3088/Memnoch
While Mike's answer is correct , I want to say that most of the discussions seem to be weak.
I believe it much more likely that Daenerys' brother was the third head.
After all, nowhere does it say that one of the heads can't die after he/
she is born.
scifi.stackexchange.com/users/947/apoorv020
In Book 4, Maester Aemon, dying, says to Sam that he thought he'd be the third.
scifi.stackexchange.com/users/1611/Theodosia
Spoiler:
It has been discussed.
The most popular contenders for the third head are: 1.
Tyrion (he has a fascination with dragons, also there is a theory that Tyrion may unknowingly be the son of Aerys and Joanna) 2.
Bran (he's a powerful warg and maybe dragons can be warged, also he has a connection with 'flying') 3.
Aegon (though there is some speculation that he may not actually be Aegon Targaryen)
scifi.stackexchange.com/users/13175/Lysa
In the series  A Song of Ice and Fire , dragons always occur in groups of three - The three conquering dragons, the three eggs etc.  Somewhere, it's also implied that there should be three dragon-controllers (I forgot where, but this is what I remember).    So if Danaerys is one, Jon the second.   Who is the third head? Has there been any discussions about this somewhere?
Who is the third head of the dragon?
scifi.stackexchange.com
<a-song-of-ice-and-fire>
scifi.stackexchange.com/questions/3011
scifi.stackexchange.com/users/947/apoorv020
Suggestions for the third dragon head include Tyrion and Daenerys' brother. Possible contenders for the third head of the dragon are: Tyrion, Aegon Targaryen and Daenery's brother.
Possible contenders for the third head of the dragon are: Tyrion, Aegon Targaryen and Daenery's brother.