[Spoilers Extended] Theory: Dany is not a Targaryen

Daenerys is not a Targaryen, she is Ashara Dayne's daughter.

Clues:

Dany is said to be the most beautiful woman in the world, as Ashara Dayne was in her time.

The Daynes have Targaryen features, explaining Dany's hair and eyes.

Daenerys misremembers the House with the red door and the lemon trees as being in Braavos when in fact they were in Dorne (where the Dayne's Starfall is located).

Switching babies is a common theme in ASOIAF. It is theorized that something like this may have happened when Ned visited Ashara and she subsequently went missing.

Dany's current closest advisor is Barristan Selmy, a man known to have been in love with Ashara Dayne. He has now, unknowingly, pledged his life to serving her daughter. He can reveal crucial information about Ashara to Daenerys at any time. In fact, he has already provided some exposition about Ashara to Dany in a chapter of Dance.

You guys are gonna hate me for this one but look at her name: Daenerys. The first five letters of her name spell "Daene". Could this be a hint from George?

Finally, if she is Ashara Dayne's daughter, it reveals that this seemingly forgotten family that was crucial to the backstory of Ice and Fire has been significant all along.