In Japan, like with several other Zelda games, Majora's Mask keeps track of a save count which is displayed on the file selection screen. The nature of saving in Majora's Mask is different though; you don't save when you run out of hearts or from a subscreen command, but when you play the Song of Time to reset the clock. (Suspend saves from striking Owl Statues is not supported in the Japanese version of the game.)

This fact, that the game effectively gives you a record of how many times you've reset the clock, makes a little more obvious an interesting fact about the game: you only need reset the clock once in order to win.

The first time through the three days you're stuck in town, and are heavily scripted in any case. It is possible to obtain three Heart Pieces during that time (four with glitches). It's good to get everything that you can possibly do in the initial three days done during that time, as you won't have a huge amount of time to dally later.

After you get the Ocarina and go back in time the first time, the race begins. You can perform every task necessary to win the game within the next three day cycle. That includes picking up every essential mask (you won't be able to get them all), including the four transformation masks, fulfilling every required character quest, and completing all four dungeons with bosses.

There's not a lot of time to spare. It goes without saying that you have to slow down the clock with the Reversed Song of Time. You have to get Epona before 6 P.M. on the first day. You have to carefully manage many timed quests so that they don't interfere with each other.

Yes, people have done this. Here's a list of what has to be done to win, which makes it seem easier than it is.

EDIT: Sorry guys, I coded the link to the list as an HTML tag out of habit.