As simple as it gets and still works
This LED flasher occurred to me while reading about negative resistance in transistors. It was reported that Leona Esaki, who was at Sony at the time, had been surprised to see a negative resistance region while investigating production defects in transistors which lead to his thesis work on the Esaki or Tunnel Diode in 1957, which eventually lead to his receiving the Nobel Prize in 1972.
In this implementation, a common NPN transistor is used. In the circuit, a 1k resistor charged the 330 uf capacitor until the voltage became large enough to get the emitter-base junction to avalanche. In the oscilloscope image, it can be seen that the peak voltage (yellow trace) was a little bit less than 9 volts. At this point transistor turned on quickly and partially discharged the 330 uf capacitor through the LED and the 100 Ohm current limiting resistor. The current wavform, which is the voltage drop across the 100 Ohm resistor, is shown in the blue trace on the scope image. Peak current was 26 milliamps, and the transistor continued to discharge the capacitor until conduction suddenly ceased at 6 milliamps (Many thanks to Luke in Australia for pointing out the correct current). After the transistor stopped conducting, the capacitor began charging again, thus starting a new cycle.
Oscilloscope image, showing the voltage across the 330 uf capacitor in yellow
know that I used a 10X probe). Thanks to a fellow named Luke in Australia for pointing this out.
If the resistor that charges the capacitor is too low in value (or if the power supply voltage is too high), the current through the transistor will not become low enough for the transistor to turn off. If the resistor that charges the capacitor is too high in value (or the power supply voltage is too low), the capacitor will not be able to charge to a high enough voltage to enable the transistor to turn on. This is because the transistor draws as small amount of current before switching on.
For more detail: Simplest LED Flasher Circuit using microcontroller