The question appears to be unnecessary.  The obvious answer is, ‘God’!  But the disciples were not really asking that question at all.  As the story continues it is clear that they meant, ‘Which of us will be the greatest in Your government, Jesus, when You set up Your kingdom in Israel?’
 
That theme continues in Matthew 20:20-28 where the mother of James and John (whose father was Zebedee) asked Jesus that her boys might be in senior positions when Jesus became King.  The indignation of the ten other disciples shows that this matter was not far below the surface.  Perhaps they thought that Jesus had already set up Peter, James and John for such roles because He selected them only to be involved at the raising of Jairus' daughter and the transfiguration (and later at Gethsemane).
 
The trainee apostles needed to understand the answer to this question because proud ambition would become a major hazard in the Early Church (Philippians 1:17; Philippians 2:3; James 3:16).  Power-hunger was a major problem for Simon the Sorcerer (Acts 8:9-25) and false prophets such as Bar-Jesus (Acts 13:6-12).  In the Old Testament, Baruch, Jeremiah’s secretary, was weary of suffering along with Jeremiah and wanted something easier or better … so God said, “Should you then seek great things for yourself? Do not seek them. For I will bring disaster on all people, declares the Lord, but wherever you go I will let you escape with your life.”  (Jeremiah 45:5). 

The issue continues to riddle the church.  Little people who crave importance use the church to glorify themselves.  Big people who have bigger ambitions may try to create a spiritual empire.  And between the extremes are a myriad of people who gain a thrill from manipulating the church from behind the scenes.  But only God is the greatest, because He is the Lord.  Compared to Him, nobody is great.  And yet there is something in most of us that long for greatness, recognition and glory.  However, if we seek it for ourselves we will lose everything (Matthew 16:26); but if we seek His kingdom and His righteousness first, then He will give us everything else we need (Matthew 6:33).