My first dive without a DM was dive number 5, or the first one out of my class. I went off a weekend or two after the class with my buddy and dove to 80 feet... Within 20 dives, I'd been to 135' and within 50 I was diving with doubles preparing for a cavern and cave course...
I've talked to some people who just don't feel comfortable diving without a DM. I don't understand that. But, I'm not the sort of person who needs someone holding his hand. I like to be independent, and value independence. The four times I have been with a DM, I felt limited by their presence.
My girlfriend is in between me and the "can't dive without a DM" people. She's got 9 dives now, and she wouldn't dive without me. Even though I'm still in DM training, she trusts me more than any other DM. She's feeling new and nervous and scared. But, she's also pretty independent, so I'm confident soon she'll be diving with other people if she wanted to. But 135'? No way. Doubles/cavern/anything technical? No way. So every person is different.
But I agree with the people who say "an open water class should train you to dive independently of a dive master or an instructor, and if you don't feel comfortable diving without one, you should probably see if there is a problem to address in your training, rather than covering up the symptoms by diving with a DM or instructor." And if at the end of the day you decide you can't live without that sheltering hand, then so be it. As long as you are having fun.