The NAB translates Mt 23:11 as, "The greatest among you must be your servant." The Douay-Rheims has it, "He that is the greatest among you shall be your servant."
The NAB's "must be" and the DR's "shall be" translate the Greek "estai," which as someone who knows no Greek I think simply means "will be." The verse may be an instruction to the greatest among us -- "Hey, great people! Be a servant to others!" -- but it could also be read as simply a statement of fact: as God measures it, greatness requires service.
What is it about greatness that requires service? I'll suggest this:
God alone is great in Himself. Humans are only great relative to each other, and the fallen human way (Jesus associated it with Gentiles) of expressing greatness is to lord it over others. The more a great man plays the lord, the closer he gets to bumping his head against the ceiling of playing the Lord, and we know how well that works out for humans.
What I mean is, we cannot become like gods by following the way of greatness. None of us is all that great.
But we can become like God by following the way of service -- specifically, the way of the greatest love, of laying down our lives for our friends. We know we can become like God in this way because we know this is the way of Jesus, Who is both God and man.
Once we become like God in this way, what do you know, we've become like Him in His greatness as well, because He will come and dwell with us, and we with Him, and He will acknowledge us as His children forever.