#Java 7, 47 bytes

    int c(double n){return n>1?1+c(Math.log(n)):0;}

**Ungolfed & test cases:**

[Try it here.](https://ideone.com/emhVB8)

    class Main{
      static int c(double n){
        return n > 1
                 ? 1 + c(Math.log(n))
                 : 0;
      }
    
      public static void main(String[] a){
        System.out.println(c(0));
        System.out.println(c(1));
        System.out.println(c(2));
        System.out.println(c(3));
        System.out.println(c(4));
        System.out.println(c(15));
        System.out.println(c(16));
        System.out.println(c(3814279));
        System.out.println(c(3814280));
      }
    }

**Output:**

    0
    0
    1
    2
    2
    2
    3
    3
    4