Why a fountain or waterfall never fills

Lets use the example below. Water flows from a hidden source into the top pool, from the top pool to the middle pool, from the middle pool to the bottom pool, but yet it never overflows. The answer is simple, the "hidden source"  is the bottom pool. Water is pumped from the bottom pool to the outlet at the top. As the top pool "fills" so the bottom pool "empties" which means there is room for the water coming in from the middle pool, which in turn is filled from the top pool.

If it rains too much then yes the pools can overflow, but most fountains have an overflow pipe.