The only condition for free fall as you said is that the motion of the body should be only under the influence of gravity alone. There should not be any effect of other forces like air resistance, viscous drag etc. The condition depends on the property of the material under free fall. For example, if the body has a certain mass as well as charged, it causes a deflection due to Earth's magnetic field. Also the Coriolis effect due to Earth's rotation (when dropped from space to earth) comes into play when the object falls from a much higher altitude.
For example, it's the air resistance that causes a phenomenon called terminal velocity. But this happens if the object is coming from a much higher altitude. At certain point during the free fall, the downward force of gravity on the object is balanced by the force of air resistance and then there will be no net force and the object falls with a constant velocity. This is why we are not get killed when rail drops fall on our head. Rain drops are under free fall. they start from a height of about $15km$. If there are no air resistance the velocity with which it reaches our head will be about $542 m/s$. That's a speed of $1951 km/hr$. Our head will be scattered apart.
So the presence of any other forces has a considerable impact on a body under free fall, especially when dropped from a much higher altitude.