Does president Obama have a stutter...?

I don't think it's an actual stutter because he doesn't do it all the time. It's an affectation that some speakers like to employ when speaking extemporaneously. They start talking before they know what they're going to say and they use it to either keep others from interrupting them or to make it appear like they have a ready answer when they actually don't. William F. Buckley Jr. was a master of this form.