As Brandon said above, the reason is that since one battery is of a greater voltage than the other, they do not entirely neutralize one another to the point where there is no current. To demonstrate this a bit more rigorously, consider the current produced by each battery. This analysis is possible through the principle of superposition:
The current produced by the 9V battery can be calculated by simply removing the 3V battery altogether, and calculating the loop voltage with only the 9V battery in place. Applying some circuit analysis to the reduced circuit, the voltage produced by the 9V source (flowing clockwise) is $1 A$.
Doing the same for the 3V battery, and removing the 9V, the current produced by the 3V battery is calculated to produce $-1/3A$ in the clockwise direction (it is actually producing a $1/3A$ current in the opposite direction, because it is oriented in the reverse direction of the 9V battery).
So because these two currents are not equal and opposite, no neutralization of current occurs in the loop.