The weight of each brick is ##mg##. You are confusing weight with normal force.
Take the middle brick as an example. There are three forces acting on it. There's the downward weight force of magnitude ##mg##, a downward normal force of magnitude ##mg## due to contact with the brick above, and an upward normal force of magnitude ##2mg## due to contact with the lower brick.
And therefore, of course, the net force on the middle brick is zero. Indeed, the net force on each brick zero, as is the net force on the entire collection of bricks.
Edit: Draw three well-separated free body diagrams, one of each brick. Identify each Third Law pair of equal-but-opposite forces. The mental effort required will benefit you greatly.