Due to the great interest in my riddle, and the overwhelming number of responses (zero), here is the answer.
If the voltage is constant, you can only complete a circuit with a wire for the electrons to leave. But suppose you had a voltage that was steadily increasing (ramping up). Then the incoming electrons have more potential than those already there, and will keep squeezing them and pushing themselves in, even thought there is nowhere for the others to go. One could theoretically light a light this way.
This is called capacitance, and capacitors are an important circuit component. They are basically a sandwich of two conductors and an insulator in between. Despite the fact that direct current (DC) cannot get through, alternating current (AC) can effectively flow in the circuit. While our homes do use AC, the capacitance of a light ciircuit is too low to create enough current to light the light at 60 Hertz. A much higher frequency, or a bigger capacitance would be needed.
For example (DO NOT TRY THIS) if one would connect a big metal plate to the dangling end of the light circuit, it would probably light up to some extent, because the plate makes room for more electrons to enter before they feel squished. They can spread out (higher capacitance).
I believe most touch screens like on a GPS work through capacitance. You can influence a circuit even though your finger is insulated via the glass.