What's the digit sum of $4444^{4444}$?

Question

For a natural number $n$ say that $d(n)$ is the sum of the digits of $n$ (in base $10$). Then what is the value of $$d(d(d(4444^{4444}))) ?$$ I have been trying with modular arithmetic, but can't do it.

Answer 1

Note that the sum of digits function is essentially a logarithm-for a reasonable mix of digits $d(n)\approx 4.5 \log_{10}(n)$ Also, the sum of digits maintains the value $\pmod 9$. So if you can compute $4444^{4444} \pmod 9$, then convince yourself that $d(d(d(4444^{4444}))) \lt 10$ you are home.