Let $f$ be the function that associates with a positive integer the sum of its digits when it is written in base 10. For example, $f(537)=15$. Calculate $f\left(f\left(f\left(4444^{4444}\right)\right)\right)$.
Let $n \in \mathbb{N}$. We notice that $f(n) \equiv n[9]$. With $n=4444^{4444}$ and using an easy induction, we get $f(f(f(n))) \equiv 4444^{4444} \equiv 7^{4444}[9]$. Since $7^{3} \equiv 1[9]$, we have $7^{4444} \equiv\left(7^{3}\right)^{1481} \times 7 \equiv 1^{1481} \times 7 \equiv 7[9]$.