First I was confused about lighting a LED without power...
Then I thought it's probably some air filter with a fan motor, and plenty of power while plugged in, but often the thing is in storage or in an attic without any power.
Correct?
Any small uC with built in EEprom will do, (or add some external EEprom, or indeed FRAM, which does not wear out, but is more then you need, but if the price is right, why not.).
You don't need an external "realtime clock" circuit, nor even a crystal to run the uC.
Running from the internal RC oscillator you usually get a clock accurate within a few percent, which is good enough for this.
There are some tricks to getting it to work reliably.
The simplest is indeed to periodically write the elapsed time to EEprom. But don't write raw data to the EEprom. Add some checksumming feature, and use multiple copies, and write some code to compare the lapsed time with the data stored in EEprom.
For example:
Use a structure for a 32 bit counter and a 16 bit checksum.
Make 10 copies of that structure in EEprom.
Increment one of the structures (and checksum) each 6 minutes, so round robin every hour (this also reduces EEprom wear)
Then, every time your gadget starts, the first thing the uC does is read all 10 structures, and figures out which one was last updated, and continues counting from there.
This simple algorithm fails if your gadged is usually turned on for less then 10 minutes at a time. So figure out a suitable interval.
You can try to do some complicated guesswork if the checksum fails, but this gets very difficult very fast. symply ignoring any counter with a failed checksum is easy and plenty good, especially because you have so many copies.
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A more complicated setup monitors the power supply voltage, and writes intermediate counter values to EEprom at the moment it detects a dip in the power supply voltage.
A moderately sized Elco is plenty to power the uC long enough to write some data to EEprom.
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As feature creep you can add a serial port that periodically outputs a string of text with data.
Version number, github link, your name number of runtime hours, how many times the unit was plugged in, anything else you can think of.
I would not recommend the ATTINY84.
If you want to use any of the AVR microntrollers, then just grab an ATMEGA328, or any other of the Mega's.
Those former Atmel controllers have lots of gotcha's and incompatibilities between different variants, and if you also start mixing Mega's and Tiny's it only gets worse. Often the Tinies don't even have a decent serial port, I2C is handled very differently from the Mega's and that's the start of a long list.
I had my final wakup call when I started with one of the 32 bit ARM cortex controllers. Started reading a datasheet about the workings of their timer peripheral, and then it had a small note: There are seven of these in this uC. If an AVR controller has 4 timers, it's likely 3 of them are different.
Price difference between Mega328 and a Tiny84 is completely insignificant for hobbyists. While the hours of figuring out the incompatibilities is not.