This is my entry to the contest. The OP creates a triangle signal with the help of a capacitor. This signal is being used to power the LED. If you want adjustable fading speed you can replace R4 with a potentiometer. The resistor R5 and the diode D1 are optional. They are being used to even out the rise and fall times of the triangle wave. 

Youtube Link:  https://youtu.be/fKUFsLTRtvE
Schematics: https://imgur.com/a/GFH0F