What is the average climb gradient of an aircraft during the takeoff and early phase of a departure procedure?
The upper limit is a function of the airplane's performance at rated climb power at the normal speed used, modified somewhat by the wind (the gradient is based on ground speed).  What actually happens on a departure is simply whatever numbers that airplane's performance produces using the standard speed and power setting for that airplane. 

In a jet you are generally going somewhere around 200 kt once the flaps are up, then accelerate to 230-250 kt once past 3000 ft up to 10000 ft.

From the table below, assuming no wind, an airliner climbing at 3000 FPM at 200 kt will have a gradient of roughly 1000 ft/nm between the initial acceleration altitude (say 1000 ft) and 3000 ft: