Solution for "f(f(f(f(f(f(f(n)))))))"

The title to this problem was a hint that there exists a function f which, when repeatedly called on a number, gives results like what was shown. The inputs and the hint text indicate that there must be something involving Base-2 (binary numbers) and English.

The secret function f is the sum of the amount of letters to spell out the number in English using base-2. For example, 11 is 13 since 11 is spelled "one-zero-one-one", which has 13 letters.

Using this, we can figure that 16383 will be 42.