I am looking for a DC circuit that outputs a constant current level even if the load resistance changes. The power source would be a 9v battery.

 

LM134 and similar will provide up to 10mA. Just Google "current regulator" or "constant current source" for others.

 

How much current? How accurate? Is there a cost constraint? Size constraint?

 

A traditional approach is to have an op-amp watch the voltage across a low-ohms shunt resistor, and compare it to a reference voltage. The voltage is proportional to current flowing through the shunt (and your load). The op-amp then controls the voltage on the base of a transistor, which in turn regulates current to the load. The current is adjustable by changing the reference voltage.