Simplest circuit to reliably produce two voltage peaks, then nothing
The simplest circuit, I believe, would be to use a microcontroller.

Many microcontrollers have a built-in oscillator and will run when powered up, without any external components.

This would of course require some basic programming and may not be the simplest solution, depending on your situation.

Personally, my approach for something like this would be an ATtiny10 or PIC10F320. With a ZIF programmer it'd be programmed and ready to go in less time than it would take me to breadboard a 555.

To change the timing I would either add a potentiometer (both of the above have ADCs) or simply socket the IC so I can easily pull it out and reprogram it.