Divisor reduction

Python, 49 bytes

f=lambda n:n and-~f(sum(n%~x<0for x in range(n)))

orlp helped save a byte! And Sp3000 saved two more. Thanks!

C, 52 bytes

g,o;l(f){for(g=o=f;o;f%o--||--g);return f?1+l(g):0;}

Pyth, 10 bytes

tl.ulf%NTS

Test suite.

Explanation

tl.ulf%NTS
tl.ulf%NTSNQ  implicit variables at the end
           Q  obtain the input number
  .u      N   repeat the following until result no longer unique:
         S        generate range from 1 to N
     f            filter for:
      %NT             T in that range, which N%T is truthy (not zero)
    l             length of that list
                  that means, we found the number of "non-divisors" of N
tl            number of iterations, minus 1.