Let $\tau:\mathbb N\to\mathbb N$ be the function that counts the number of digits of an nonnegative integer, i.e. $\tau(x)$ is the number of digits of $x$ in base 10. For example $\tau(5)=1$, $\tau(0217)=3$ and $\tau(10^n)=n+1$ ($n\in\mathbb N\cup\{0\}$)
Let $(u_n)$ be the sequence defined by $u_1=2^{3^{4^{5^{6^{7^{8^{9^{10}}}}}}}}$ and for all $n\in\mathbb N$, $u_{n+1}=\tau(u_n)$ .
Find : $n_0=\min\{n\in\mathbb N : u_n=1\}$
I've invented this, but I have no idea how to find it.
Well, maybe just one idea : $\tau(x)=[\log_{10} x]+1$