[Hugh R]

From Straightdope.com that Moses posted.

a digital clock, nine is the greatest interval obtainable by advancing some sort of "snooze counter" on the ones column. But why mess with the ones column? Why not put the snooze counter on the tens column and advance that by one?

This one almost makes the most sense. I remember in college I built a digital clock from scratch for an engineering class. What I remember is that the hour, minute and in my case seconds displays all work independently of each other. In other words, going from 9:59 to 10:00, the 10 changes without info from the 5 or 9 display. So the 9 minutes is the longest time period that you can get.