Nice one ThunderStone.  As you can see the default number of bounces on a reflection is 6 (see the black plane marking the 6th reflection in TS's doors render).  This number can be increased within the render options menu.  

Simply put the camera in front of the first mirror instead of behind it.  The camera is invisible in the scene.  If you are wanting to have some scenery behind the first mirror, why not render a separate scene and mask out the mirror portion there and layer it in post work.