It’s common knowledge that you need a ‘closed circuit’- an unbroken, continuous, conducting path- for an electric current to flow. If you are using a battery, this usually means an unbroken path from the positive terminal of the battery, through an LED (or whatever device you are running), all the way to the negative terminal of the battery.
But the closed loop between the terminals of the battery is strictly not necessary. What is important is that an electric current needs to flow through the LED, and for this all that is required is that the LED is connected between two points at different electrostatic potentials. The terminals of a battery contain static charges, and one could theoretically draw a small current for a small duration if we connected an LED between one terminal of the battery and a neutral object. The neutral object will act as a source or sink of electrons, depending on whether we are connecting it to the positive or negative terminal respectively. But for the chemical reactions in the battery to continue happening to provide a continuous current, the other terminal also needs to be operating (this is something that needs discussion, but I’ll do it in another post).
To test this out, I connected the positive lead of the LED to the positive terminal of a 9V battery, and held the negative lead with my fingers (myself being the neutral body). Obviously the LED didn’t light up. But then I connected the negative terminal of the battery to the earthing in an AC mains socket, so that it can act as a sink for electrons from the negative terminal of the battery. And the LED lit up! Not brightly, but that’s understandable, since my body has a large resistance.
Here’s a photograph of the LED glowing when I touch its negative terminal. The second picture shows the LED when it’s off, so that you can see the difference.