Imagine a civilization which has surmounted almost all of the limitations of biology and the universe. They have become truly immortal, put behind them all their petty differences, and dug down into the core of the laws of physics and somehow figured out how to stop entropy and produce free energy. Now they must face their biggest challenge: time notation.
For the first 100,000 or so years, it's fairly easy to just keep up with the current form of time notation. Just update the code of apps to support 5- and 6-digit years at the 10,000- and 100,000-year mark.
But as the millennia turn to eons and longer, eventually writing down the full year becomes somewhat infeasible, and our civilization realizes that it must come up with a way to cope with year numbers increasingly becoming longer.
Eventually, the year passes 101000, 10100000 and more, eventually reaching into the millions and billions of digits, becoming incredibly difficult to log.
Our civilization wants to keep a well-detailed history and doesn't want to resort to things like only storing the last four or five digits of the year and just relying on people to know what era something occurs in by context.