The Official Daenerys Targaryen in the tv series Game of Thrones is played by Emilia Clarke.
Update June 2011: Emilia is a newcomer like most of the cast. I wasn't sure about her at first but she's grown on me and she's portrayed Danny quite well.
My fantasy cast for Daenerys a few years ago was Melissa George.
Daenerys Targaryan in A Song of Ice and Fire is a the daughter of King Aerys II, and her mother was Aerys's sister, Queen Rhaella. Daenerys was conceived during Robert Baratheon's rebellion that would ultimately end her family's reign. Before she was born, her pregnant mother and her brother Viserys fled the royal seat at King's Landing for the safety of Dragonstone, the family's ancient home.
Daenerys was born as a fierce storm struck Dragonstone, and so was called "Stormborn." Her mother died as a result of the labor. Shortly thereafter, Dragonstone fell to the rebel forces, and Daenerys and Viserys were smuggled away to the Free City of Braavos by a loyalist knight, Willem Darry. Ser Willem was old and sickly, but he treated Daenerys kindly and fter his death, his servants drove Daenerys and Viserys from the house. Daenerys wept as they were forced out, and the house with the red door became her symbol for the childhood she had never known.
In the Free City of Pentos, the Targaryens were the guests of Illyrio Mopatis, a rich and powerful magister. Illyrio and Viserys arranged to marry the thirteen-year-old Daenerys to Khal Drogo, a great leader among the Dothraki people, who are nomadic horse lords. Viserys hoped to use the thousands of men in Drogo's khalasar to conquer Westeros. Despite Daenerys' fear and apprehension, she had no choice but to accept her brother's will and wed Drogo. At her wedding, the exiled knight Jorah Mormont pledged his service to Daenerys. He would become her closest companion.
Daenerys grew to love her husband, Drogo and blossomed into a strong and confident khaleesi (queen).
Although there are hundreds of young blonde actresses that may make a good Daenerys, once Melissa George came to mind I couldn't shake it off - so throughout the book, Daenerys to me was played by Melissa. The photo above was photoshopped with Melissa's eyes colored purple.