Medium Transport; Low Wing; 2 Turbojet Eng
Air Traffic Control Fully Certified
Air Traffic Control Time Certified In Pos 1 (mon) 4
Communication Breakdown; Confusion
ATC Issue All Types; Conflict Airborne Conflict; Deviation - Altitude Excursion From Assigned Altitude; Deviation / Discrepancy - Procedural Clearance; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence
Air Traffic Control Issued New Clearance; Flight Crew Returned To Clearance
Chart Or Publication; Human Factors; Procedure
Working all the departure positions combined; early morning light traffic; weather in the area and route restrictions in effect. Aircraft X was the last departure not on vectors before Tower assigned vectors. I was on the land line when they checked in coordinating departures with DAL Tower. They departed on a heading for weather and reported altitude leaving. I missed the no altitude assigned being on the land line but expected a climb to 100 as they were on a heading. I turned them east; they were a south gate departure; to get in trail of the other southbound departure I had. The south gate was one route in trail for the weather. I then adjusted their heading to a 070 to give myself a little more space in trail. I went back to other aircraft and when I next looked at Aircraft X they were climbing through 110 right at an arrival level at 110. I turned Aircraft X immediately to a northeast heading; I think a 040; and then came back and asked what altitude they were climbing to. The pilot said FL380. I amended their altitude to 170 and eventually turned them south. Immediately cancel the climb via procedures. Foreign pilots and pilots unfamiliar with any area are apt to climb to a wrong altitude. This is just creating an extremely unsafe occurrence. Even with ATC catching wrong altitude check in or missing them there is still the chance for pilots to climb through the needed or should be assigned altitude. The safer procedure should be to just leave the altitude assignments in ATC's hands with hard altitude assignments. This is only an unsafe operation when control is given to no one.
Reporter states he missed altitude aircraft was climbing to on initial call up.  Reporter assumed aircraft was climbing to one altitude but pilot reported climbing to a higher altitude.