Hi Chris,
A coil can only saturate if it has a magnetic core – saturation happens when the magnetization of the material reaches a maximum, even when we increase the applied magnetic field. An air core is saturation free – however due to the resistance of the copper wire there is a limit to the current that a coil can accept – heat released is proportional to the resistance of the coil and the square of the current. Although it depends on a few factors such as geometry and air convection, it is accepted that a 15AWG copper coil has a typical current rating around 20 amps.
Although if we accept large size and cost the rule does not apply usually there is compromise – air core has higher resistance and no saturation, magnetic core has lower resistance but saturation issues. Many people tell us that a properly dimensioned coil with an adequate core does not have listenable saturation effects before speaker damage.