The scaling in the overlap region of turbulent wall-bounded flows has long been the source of controversy, and until recently this controversy could not be addressed because measurements did not span a sufficient range of Reynolds number. Mean velocity surveys performed in a new pipe flow experiment span a very large range of Reynolds numbers, 31×103 to 35×106 (based on average velocity and diameter). Here, these experimental data are used to evaluate theories on the scaling in the overlap region. At sufficiently high Reynolds numbers, the mean velocity profile in the overlap region is found to be better represented by a log law than a power law. These results suggest a theory of complete similarity instead of incomplete similarity, contradicting the theories recently developed by Barenblatt et al.