Since the question posed 'the greatest...' - i'd like to say that that is entirely in the eyes of the beholder. After all, once one has completed a bachelor's degree in piano, they probably have a good idea of their own interests and peculiarities towards piano. Asking a 'brilliant' teacher really has nothing to do with one's personal interests. Just as with any field of endeavor - you start honing and specializing what YOU are personally interested in. Therefore, it is less about the teacher and more about what you want. Otherwise, you're wasting money. Why spend a lot of money on lessons to hear something you don't want to hear or aren't interested in.
Thibaudet decided that when he went to (Paris Conserv?) that he wanted to only play Chopin. Therefore, he probably ended up with the teacher that helped him the most with chopin. Perhaps a student of a student or something.
With these teachers, who cares if they don't take beginners (perhaps only genius child prodigies). After all, they have paid their dues and don't want to be teaching something that someone else can do just as easily. Playing piano is a succession of difficulties. First you learn to listen, make tones, read music, etc - on through college - where you learn discipline and hard work. By the time you get to master's level and beyond - the student is supposed to be the one that is self-motivated - so the goals change.
Basically, imo, the goal is to ask the right questions. So...what would i ask personally? Well, i'd ask myself if i deserved to take lessons with this person. For me it was a resounding 'no' after the first lesson (with someone who graduated from Julliard). But, piano is a bad addiction for me. Even if i could only learn one or two things from each lesson and fully adopt it into my playing each week - it would be worth the money of the lesson. If you take from someone who doesn't bring up or down the level of intensity - you could be like an autistic person trying to learn from a nasa astronaut. What's that going to get you? A lot of frustration and probably being knocked to the side of the wall a few times.
ps no discouragement to autistic people, btw, because some can play piano fantastically. it's just the method is much different. a lot of repetition.
pss sometimes not asking any question at all is good. just letting them talk after those endless pauses. perhaps slowing down is the students mental 'best-friend.' i was so extremely anxious and nervous i'd forget to breathe sometimes. just remember, these people aren't gods. they're just regular people (we'd like to think).