I remember getting about 5 mins off of a 4400 uF capacitor array. After that the clock lost time and I had to double-flick it to start it.

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That's interesting . So a 10F cap should last nearly 8 days. That's actually not great.

How did you test this? Did you install a charged cap instead of the battery?

PS: if I did my calculations right, a 300F cap would last nearly 34 weeks, still nothing compared to a battery.