Once, Twice, and Thrice

Inspired by a post by Grumpy Old Ken:  Twice for two times can hold its own when tied to once as in "Once burned, twice shy" or "I did that once or twice".  One time is sometimes used for once but it would be awkard to start a story "One time upon a time."  By the way, why is it "upon a time" instead of "at a time" or "during a time"?  And couldn't "upon a time" be dropped?  Not as verbose as "at that point in time" for "then".  Finally, is time after time two times?  Minds with too much time on their hands want to know.  I always keep time on my hands.  Where do you keep yours?