If the mass of an object is 1 kg, what is the range of possible accelerations?

The range of possible accelerations of a body is dependent on two things: its mass and applied force. According to Newton's Second Law of Motion, force, mass and acceleration are related as follows:

F = m x a

where, F is the applied force, m is the mass of the object and a is its acceleration. If we know the mass of the object (say 1 kg, as in given case), we need to know the applied force to determine the possible acceleration, as a ratio of applied force and mass of object (a = F/m). For example, say an external force of 10 N is applied on the body, then its acceleration would be 10 m/s^2 (= 10/1). However, if we applied a force of 10 kN, the acceleration would be 10,000 m/s^2.

Thus, without knowing the applied force, we cannot determine the acceleration. And the range of acceleration would depend on applied force. 

Hope this helps.