I'm making a Tamagotchi, which requires that I keep track of the time since I last used my device, so that its hunger grows and whatnot. I don't own an RTC or anything of the sort, only an Arduino Pro Mini. This doesn't have to be accurate in any way whatsoever, just enough so that a number increases or depletes over time.
How would I go about doing this with minimal battery usage?
A RTC will be the best solution, or a 32kHz (watch) crystal for the T2 timer/counter.
Power consumption depends on Vcc, and also is almost proportional to the CPU clock frequency. The oscillator frequency (16MHz) can be divided by the clock prescaler, so that the controller will not run faster than required for your project.
When you select a sleep mode during inactive periods, an Arduino clock must keep running for counting cycles, and the controller has to wake up with every counter overflow. This can be achieved with power-save mode, where T2 continues running. Entering sleep modes have been discussed frequently in this forum.
Nick Gammon's power saving tutorial may be useful
...R
Just a thought, I believe one of the small siblings of the Uno, like a Nano or ProMini, can operate on microamps when in sleep mode. What if you use the watcgdog timer to wake up every eight seconds and iust increment a counter. Then time elapsed can be derived as 8*counter sec.