Since $3 \times 5 \times 7=105$, then the greatest possible value of $n$ is at least 105. For the product of three numbers to be positive, either all three numbers are positive or one number is positive and two numbers are negative. If all three numbers are positive, the greatest possible value of $n$ is $3 \times 5 \times 7=105$. If one number is positive and two numbers are negative, the greatest possible value of $n$ is $7 \times(-4) \times(-6)=7 \times 24=168$. Combining the two cases, we see that the greatest possible value of $n$ is 168.