I love the classic Doctor Who series. I'm not a big fan of the new series, but it's occasionally fun. To me, the remarkable thing about both old and new is, despite the fact the character of the Doctor has been played by very different actors, the personality type has remained constant throughout. He's ENTP, no matter who is playing him.
E - never wants to be alone...yes, he may say that he does sometimes, often grumbling about his companions, but his behavior shows he never really wants to travel alone, or be alone for any length of time. He is clearly energized by being around other beings, humans or otherwise.
N - though he certainly uses his senses, his primary mode of operation is N. He's always looking to future possibilities and trying to anticipate the next move of villains and friends alike, based on what he gathers about them.
T - he may have his emotional outbursts, but his basis is definitely very logical. He dismisses people/creatures who base decisions on emotions or illogical traditions or superstitions.
P - He's the classic "mad professor" in his style. Very haphazard, doesn't like to follow schedules or set plans, likes to jump into new experiences and environments and just let things happen. His home, the TARDIS, is extremely disorganized, and he spends lots of time hunting for tools and other things he's collected because he has tossed them somewhere without much care as to where they've gone, just knowing he'll find them again eventually.
The companions, well, there's been so many over the years, and since I only just recently got into MBTI, I haven't watched a lot of episodes with intent to figure out types. But the most recent story I watched was the 5th Doctor story, Terminus. So my guesses on his three companions at that time:
Nyssa - ISTJ (she is friendly but not particularly interested in spending time with a lot of people, somewhat reserved. She is very logical and scientific but also very traditional in her thinking, she trusts what she knows and that things she has to figure out will be done by the processes that she knows.)
Tegan - ESFP (the "mouth on legs", very action oriented, jumps into things without thinking, very hands-on, and even though she has a quick temper, actually seems quite friendly a lot of the time.)
Turlough - ESTP (It's hard to tell, since at this time, he's under the control of the Black Guardian, but he seemed to be a popular and tactical troublemaker when we first met him at the boys' school, and he is definitely focused on what action needs to be done at the moment, rather than trying to figure out people's motives or make plans.)