It might not have to survive without power.
If you power up, read a byte from eeprom, you can know what mode you were in last time around, increment that value, then power down.
Next time you power up, you read the same byte, the mode has incremented, save the value, power down...wash lather rinse repeat.
And there's probably some slick single-button power on/off trickery going on.
Do a search and you'll find a slick single-button power on/off trickery circuit here too...