Paul's statement regarding being the chief or greatest sinner must be taken in context. First, the Apostle Paul thanks God in heaven for being merciful and counting him faithful enough to minister to others fulltime (1Timothy 1:12). This mercy was given to Paul even though, at one time, he was unknowingly a "blasphemer and a persecutor and a violent person" (verse 13, HBFV throughout).
Chief Sinner
After Paul mentions how the Lord's grace abounded toward him (1Timothy 1:14) we come to the verse in question in which he calls himself the worst or greatest sinner.
This is a faithful saying, and worthy of full acceptance: that Christ Jesus came into the world to save sinners, of whom I (Paul) am chief (1Timothy 1:15).
In 1Timothy the apostle is primarily referring to the time, before his calling and conversion, when he persecuted Christians by arresting them in their homes (Acts 8:3). He made people blaspheme against God and his zeal even led to him pursuing true believers in foreign cities (26:11).
Paul also took part not only in the death of Stephen (Acts 7:58) but of many others (22:4). He sums up his life as a "sinner for God" in the letter he wrote to the church in Galatia.
For you heard of my former conduct when I (Paul) was in Judaism, how I was excessively persecuting the church of God and was destroying it (Galatians 1:13).
Yet, in spite of his zealous but deceived efforts to destroy the true church build by Jesus Christ, Paul was shown mercy as an example of God's love and patience (1Timothy 1:16).
No person lives for any length of time without committing sin. Even those who are converted, who have the Holy Spirit in them, struggle daily with the downward pulls of human nature. Paul himself commented on his own personal struggles he had to deal with even after he was converted and made an apostle of the Gospel.
Because what I am working out myself, I do not know. For what I do not desire to do, this I do; moreover, what I hate, this is what I do . . .
For the desire to do good is present within me; but how to work out that which is good, I do not find. For the good that I desire to do, I am not doing; but the evil that I do not desire to do, this I am doing (Romans 7:15, 18, 19).
The Solution
The solution Paul discovered to the conflict between good and evil he experienced every day was to trust that God would ultimately save him (Romans 7:24 - 25).
Although Paul felt he was the greatest sinner before being called during his famous trip to Damascus (see Acts 9:3 - 6, 16 - 18), the Bible offers examples of those who did far more evil.
King Ahab and his wife Jezebel helped strengthen the grip of idolatry of Israel that ultimately led to the whole nation going into captivity. Doeg not only killed Israel's High Priest but also other priests he could find and went to the city the High Priest lived in and killed as many people as he could (1Samuel 22:18 - 19)! There are many other Biblical examples of evil people that show that Paul was far from the greatest sinner!