What exactly is the requirement (in terms of saturation) for a reaction to be at equilibrium? A saturated solution is a solution that contains the max amount of solute able to be dissolved. However, can't a solution still be at equilibrium if you add more solute to a saturated solution as long as it doesn't reach supersaturation? What exactly is the requirement (in terms of saturation) for a reaction to be at equilibrium?
If you add solid sodium chloride to a saturated sodium chloride solution, it is still at equilibrium. If you increase the sodium ion concentration or the chloride ion concentration (by e.g. adding a high concentration of NaOH or LiCl), then the solution is supersaturated and will not be at equilibrium.


  What exactly is the requirement (in terms of saturation) for a reaction to be at equilibrium?


A reaction is at equilibrium when the reaction quotient is equal to the equilibrium constant. For a precipitation reaction, the solid species is not part of the equilibrium constant expression, so adding more solid does not change the situation. Changing the concentration of the ions in solution, though, will change the reaction quotient, and the reaction will no longer be at equilibrium.