Just for interest, 5 hours after I started the sketch is still running. The power is down to 3.9V. So as a proof of concept, it appears that a couple of ultracapacitors can keep your sketch running for 5 hours. This particular sketch spends all its time lighting up LEDs, which is probably a bit of a torture test, comparing to monitoring temperatures etc.The two capacitors I used cost me $US 32 (for both).Whilst they take a while to charge, I think this is acceptable. After all, you put charged batteries in as a battery backup don't you, not discharged ones. Either you could organize a more sophisticated charging circuit, or just pre-charge them.Something like this could keep a security system running for hours, even during a power outage. And if you double the number of capacitors, you double the time they will keep running. Coupled with the suggested power board, with its DC to DC converter, you could probably squeeze something like 8 or more hours of performance out of them (or 16 if you use 4 capacitors).
Bloody heck, why not just use a truck battery
I don't want to use a battery because of the extra cost and complexity ...
In the first version this will only take a second or two but later I might need up to a minute.