Are there Multi-Turn Absolute Encoders that Store the Number of Turns in Non-Volatile Manner?
Manufacturers choose different strategies about absolute encoders. Asian and Japan manufacturers tend to use kind of incremental encoder that needs an additional lithium battery at driver side to power the encoder at power outage. 

European manufacturers usually use hall sensors with magnet gears, similarly like a water counter for counting revolutions, while the disc is made of glass or it's a resolver - this is so called multiturn absolute encoder. Disc or resolver gives the position within one revolution, while the gearbox measures numbers of revolutions. Such encoders don't need power for counting position. However an EEPROM inside is a good to have, used to store reference or null position.



Encoder Gears

I had used Sick, Kuebler, Indramat,...I can confirm it does not need power. While Fanuc has an additional battery in motor driver.