$m\frac{g}{k}$ below

small

$2m\frac{g}{k}$ above

large

above

below

$m\frac{g}{k}$ above

$2m\frac{g}{k}$ below

In order for the spring to exert no restoring force at all, the oscillation would have to be so that the highest point is at or the initial, unweighted equilibrium end of the spring. In this case, as the mass passes through the point where the unstretched end of the spring would hang if unweighted, the spring would momentarily exert no force on the mass.

The new equilibrium point is a distance the initial unstretched bottom of the spring, where $m$ is the mass of the hanging object and $k$ is the spring constant.