Touch sensor with a CR2032 battery and a transistor
You could play with something like this:



simulate this circuit &ndash; Schematic created using CircuitLab
The CR2032 has enough internal resistance that the LED does not burn out.
There's a good chance M1 could be damaged by touching the gate but if you touch the (-) battery terminal first it should work for a while.
Eventually the gate leakage will cause the LED to turn on or off (by itself) after some time.