To achieve the target number $10$ using only the numbers $4$, $6$, $6$, and $9$ once each, and with the basic arithmetic operations (addition, subtraction, multiplication, and division), we can proceed with the following solution:

$9 - \left(4 \times \left(6 \div 6\right)\right) = 10$

Here's the step-by-step breakdown:

1. $6 \div 6 = 1$
2. $4 \times 1 = 4$
3. $9 - 4 = 5$

This yields $5$, so it doesn't directly achieve the target of 10 based on