lol Not many actually, I did about 20 or so. I wanted to do all of the missions but got turned off after the gauntlet missions (1000 enemies?!). I'm definitely going to go back to it later, after I... purchase the UMD.
But for now, I just want to play FF7 next.
EDIT: Sonic, I think I read somewhere that the missions add about another 50-60 hours of playtime.
EDIT: EDIT: Does the game save your mission progress when you complete the game? I didn't check.