LM317T outputs current on Adjust pin with nothing connected between adjust and Vout?
Yes, this is normal. The internal circuits in any three-terminal regulator require a certain amount of "quiescent" current to operate correctly (some more than others), and this current flows through the "adjust" or reference pin.

When used in a current regulator configuration, this quiescent current becomes the minimum output current. Even "low power" regulators (Iq &lt;&lt; 1 mA) will pass enough current to light an LED visibly.