Inspiration

$(10^n)^{10^n}=10^{n \times 10^n}$ so $n \times 10^n$ have to be lower than $10^{100}$ .The answer is $\frac{W_n(10^{100} \times (\log(2)+\log(5))}{\log(2)+\log(5)} \approx 98$ ,since $10^n$ have $n+1$ digits so the number of digits is $98+1=\boxed{\large{99}}$

Note

• The definition of $W_n(x)$ is here