Two solutions:
Let $S(n)$ denote the successor operation over the natural numbers, that is $S(n)=n+S(0)$. $S(S(S(S(S(S(0))))))=6$
Similarly using the cardinality operator, the power set operation $P(x)$, and set difference, we can write:
$6 = \big|P(P(P(P(\{n\in\mathbb N\mid n<0\})))\setminus P(P(\{n\in\mathbb N\mid n<0\}))\big|$
Now, why this question generally annoys me? Well, coming from a logic and set theory point of view I am aware that $0$ is merely a syntactic creature. The question asks something about how many ways there are to write a certain constant by a formula.
There is no specification of the language, but it is somewhat implied that the operations should be "mathematical" in their nature. I use quotation because of course one can always define $f$ to be some bizarre permutation of any set containing both $0$ and $6$ and simply write $f(0)=6$.
While the question clearly means "high-school'ish" mathematical operations, it is ill-defined, almost as the questions of the form "Guess the next number in the sequence: $a,b,c,\ldots$" - the mathematical answer is usually $42$.