Well rate of climb is dependent on the amount of power available, jet or thrust available piston, over that which is required to over come the drag---
as the engines of all planes lose power with altitude---
meaning, when the amount of power or thrust available no longer equals that which is available you end up with a situations whereby you reach certain points on the thrust altitude curve where---
you'd reach a service ceiling ---the exact definition of which depends on the certification standard depends of the service ceiling depends on the certification whether far 25 or 23 [military airplane have combat ceilings also],
but, if you really wanted to---
you could keep climbing with a 'residual rate of climb' until you reach a point where the power available is equal to the power required and climbing is no longer possible However,
you would have to maintain the best rate of climb to reach this point...
with high altitude and high speed you run into other troubles due to shock waves building up shaking the plane because the air can't escape the path of the airfoil and gets compressed, but that's a complex topic do a search for 'coffin corner' and you may find interesting stuff.