As far as I can tell.. there isn't a limit per day. I guess if you make a second account, you could continually gift each account gifts until you have enough money or whatever you want.
It might eventually tell you that you've reached your limit for the day, but I haven't encountered one yet. (sent 10 or so in a day)
You can send one gift per recipient per day. So, if you send a gift to Aunt Suzie today, you can't send her another one until tomorrow.