This is a typical problem that can be solved by the MOD-by-$9$ method.

Because $4444^{4444} < 10000^{4444} = 10^{17776}$, we find that $4444^{4444}$ has at most $17776$ digits, which means that $f(4444^{4444})$ can not be greater than $9 \times 17776 = 159984$. It follows that $f(f(4444^{4444}))$ can not be greater than $9 \times 5 = 45$. Similarly, $f(f(f(4444^{4444})))$ can not be greater than $3 + 9 = 12$.

Meanwhile, we have $$f(f(f(4444^{4444})))\equiv 4444^{4444}\pmod{9}$$ and $$4444^{4444} \equiv (-2)^{4444} = 2^{4440} \times 2^4 = 64^{740} \times 16\equiv 1^7 \times 7 \equiv 7 \pmod{9}$$

There is only one positive integer no greater than $12$ which is congruent to $7$ modulo $9$. Hence, the answer is $\boxed{7}$.