Then there arose a reasoning among them, which of them should be greatest. (Luke 9:46)
The way the scriptures talk about this incident we learn from Jesus’s response that it was kind of an egotistical thing for them to do to reason who of them should be greatest. But I suppose if we were to think about it, we’d probably see it came out of a different place than ego and then turned into ego.
I bet that it first came out of the consciousness of the amazing things Jesus did and their knowledge that He was the Messiah, the anointed one. From this they would begin to ponder the high privileged role they each filled in being so close to Him, being of that inner circle. They knew Jesus was going to be remembered and I suppose they wondered if any of them would be remembered too. This would cause them to ponder which of them deserved to be remembered for being the greatest, next to Jesus of course.
This is when Jesus tells them that the greatest will be the least, or the servant of all. I guess one shouldn’t spend time thinking about one’s legacy, but rather concentrating on finding ways to be of service to others.