pouring concrete right next to asphalt driveway
If, when pouring the concrete right up against the asphalt, you end up with parts of the concrete that taper or feather out to less than say about 1.5 -> 2.0 inches thick you will have trouble keeping the concrete in one piece. The thin parts are likely to fracture and break off. 

I suspect that if you rented a concrete type saw and cut off the asphalt in a nice square edge that you could then pour the concrete right up to it without a form. You will want to make sure the concrete is at least 3.5 -> 4.0 inches thick to make sure it stands up to be driven upon and changes of weather through winter seasons (if you live in a place where it freezes in winter).