Saving integer data after power shutoff

For your simple application, you don't need a library. Each time you write, search the entire EEPROM for the largest number. Write the new number (largest number + 1) so as to overwrite any other number. Edit - the next consecutive number (in order to keep the location "rolling"). If you use 32 bit numbers, that gives you a count of billions.

The only catch - at the beginning you have to fill it with zeroes. Because, an erased EEPROM is full of 1's, i.e. full of bytes of 0xFF.