After some trials and errors, I've settled to this system:

3300µF was what I had easy access to, I wanted to make a ring oscillator so I looked for 3 identical high-valued electrolytic capacitors and they looked nice.

I get about 12Hz. The frequency is affected by the series resistor, which I had to reduce in order to keep a stable oscillation because it doesn't start up quiclky.

I initially started with a safe value of 47 Ohms because the coil is 35 Ohms but it didn't work well. It was a bit better at 22 but 10 Ohms keeps oscillating at 4V. The inrush current is in the order of 400mA... (coil dropped below 1V, Vcc=5V, I=(5-1)/10=0.4A)

Maybe I should add a lamp in series to keep it running ?

Anyway, with 10 Ohms, the charging pulse lasts about 10ms, with 65ms of discharge.

That's about 1/75ms=13Hz.

Zooming in the charging pulse, I can see strong ringing...

No wonder my frequency meter was confused !

More action on youtube:

20161107: Oh yeah, the inrush current is 4× the rated 100mA current... No wonder it doesn't work well.

I'm about to make a longer ring oscillator usign PBRL to test the endurance, this will limit the current and check the speed up to 50Hz...