*Edit - ok, "straight" answer - to be honest, probably a little - in '66. Surely less than most anyone else on earth in that same position, which no one else ever would have, could have, has, or probably will get into. John Lennon called that arrogant attitude in his own life his "Elvis" period, btw.
He is actually down to earth, but has very good reasons for distancing himself from fans and the press. He has always been about the music - never about his "laurels". He has always respected other musicians opinions more than fans, or especially "the media". His greatest honors at this time were the admiration he garnered from such musicians as The Beatles, Jimi Hendrix, and others of them thar' ilks. Not because of their success, but because of the honesty in their music and persons.
Just before going on The Ed Sullivan Shoe, CBS executives told him he couldn't play "The John Birch Society Paranoid Blues", which had been rehearsed for like two weeks and all - Dylan walked! Integrity, not "fame".
The '66 world tour is probably as close as he ever came to being swell-headed. He was kicking *** (the world's) and taking names!
In regards to the incident in "Don't Look Back". Donovan was a pretender who was being promoted as the "British Bob Dylan" - I can't think of any other musician who ever had the gall, and/or lack of integrity to allow themselves to be promoted that way (arbitrarily attaching themselves to another musician) and Dylan was having none of it - to his credit. Integrity, not "fame". This tour was also shortly after the famous/infamous Newport incident where he went electric, to the dismay of the folk scene that he was at the top of (I'll try to post a clip of some of that, below), and he was being widely booed by purists/phony fans (I thought of a clip of this too, if I can find it). Anyway, both hotel incidents were involving Donovan and/or his entourage. One of them tore something glass off a wall in the bathroom and threw it down in the alley and Dylan rightly got upset about that, it being his room and all. This was no doubt related to the confrontation between Dylan and Donovan that had just occurred. Pay particular attention to the weak (not bad) song that Donovan played (showing off), then Dylan taking the guitar away from him and tearing into him with his own song. I think, on a personal basis, he was a bit rough on Donovan, but there was additional supporting context. I don't see it as his finest moment, but who wouldn't want that as their worst? . . . which is what it is as far as I know.