The translation of the word "greatest" means "greater." Indeed, "greatest" is not a term Jesus ever used because it would refer to the Father. The word means "greater" and the form of the following "you" means "than you." However, it also means "higher," which is important in the meaning of the next verse, Matthew 23:12.
The word translated as "servant" is misleading because the KJV almost always translates a different Greek word, actually meaning "slave," as servant. I suspect that those hearing it heard something like our word "underling," contrasting its meaning with the "higher" above. Also notice, that it is not God that is served but people serving each other.