The Greek akribestatos (ἀκριβέστατος) is the superlative of akribes (exact, precise), meaning the most exact, strictest, or most rigorous. Paul uses it in Acts 26:5 to describe the Pharisees as the strictest party of Jewish religion.
Paul's claim establishes credentials within Judaism — his encounter with Christ was not ignorant enthusiasm but a costly conversion from the pinnacle of Jewish religious achievement. The gospel is not for the uninformed; it calls the most rigorous to bow before Christ. Paul's transformation from strictest Pharisee to apostle of grace is history's greatest testimony to the power of the risen Jesus.