Im fairly sure hes ENTX. Is very charming especially in his public speeches but one-on-one has a slightly cold, clinical air to him. Well balanced between thinking and feeling but I would edge more towards T.
Leads with a judging function rather than a perceiving one. Everything he says has the feel of finality, decisiveness, a judgment made. He has the wicked sense of humour I would associate with an Ne, but his jokes seem prepared and are delivered in a deliberate fashion, rather than having that impulsive, off-the cuff style of an Ne. So I go TeNi: ENTJ.
Coupla fun videos to chew over if you are so inclined. I welcome debate on this matter! Please tell me how I am wrong! I would be most easily persuaded towards Fe - especially since that seems to be the consensus on this thread and I am inclined to value public opinion.
[Edit] Obama's slogan of "Yes we can!" and his book "Audacity of Hope", and general focus towards change and the future, are heavily indicative of an Intuitive. Possibilities over actualities.