1400 on my original one, about 1250 on my second game, around 200 on my third game, and about 150 or so on my latest one. There's also a few specialty games I've messed with, and a few lost games. So maybe 3500, or so.
It is possible to have 65535 days.
That's about 180 years, excluding leap years.
I gamesharked the day to 65535 (hex value: FFFF) to see what happened, but nothing special occured. However, the day changed to 130xxx or some value like that after the race was over.
I get bored after about 1000 days or so, and thus, it's time for a new game.