That's a tougher question than you might imagine.
Some cells turn over each day.
Others are replaced in 7-10 year cycles. So you may have been born in the Big Band or Rock and Roll era, but much of you is really part of today's Hip Hop generation. You are probably as old as the neurons and skeletal muscle cells you were born with and are still around. These cells do not completely turn over and can stay with us throughout our lives. Still, molecules within these cells are replaced, so even the age of these old cells may not be our true age.