Let $G=10^{10^{100}}$ (a.k.a. a googolplex). Then \[\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\] can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Determine the sum of the digits of $m+n$.

[i]Proposed by Yannick Yao[/i]
1. First, we need to simplify the expression \(\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\) where \(G = 10^{10^{100}}\).

2. Start by evaluating \(\log_{10} G\):
   \[
   \log_{10} G = \log_{10} (10^{10^{100}}) = 10^{100}
   \]

3. Next, evaluate \(\log_{\left(\log_{10} G\right)} G\):
   \[
   \log_{10^{100}} G = \log_{10^{100}} (10^{10^{100}})
   \]
   Using the change of base formula, \(\log_b a = \frac{\log_k a}{\log_k b}\), we get:
   \[
   \log_{10^{100}} (10^{10^{100}}) = \frac{\log_{10} (10^{10^{100}})}{\log_{10} (10^{100})} = \frac{10^{100}}{100} = 10^{98}
   \]

4. Now, evaluate \(\log_{\left(\log_{10^{100}} G\right)} G\):
   \[
   \log_{10^{98}} G = \log_{10^{98}} (10^{10^{100}})
   \]
   Again, using the change of base formula:
   \[
   \log_{10^{98}} (10^{10^{100}}) = \frac{\log_{10} (10^{10^{100}})}{\log_{10} (10^{98})} = \frac{10^{100}}{98}
   \]

5. The expression \(\log_{\left(\log_{\left(\log_{10} G\right)} G\right)} G\) simplifies to:
   \[
   \log_{10^{98}} (10^{10^{100}}) = \frac{10^{100}}{98}
   \]

6. We need to express \(\frac{10^{100}}{98}\) in the form \(\frac{m}{n}\) where \(m\) and \(n\) are relatively prime. Since \(10^{100}\) and \(98\) share no common factors (as \(10^{100}\) is a power of 10 and 98 is \(2 \times 7^2\)), they are already relatively prime.

7. Therefore, \(m = 10^{100}\) and \(n = 98\). We need to find the sum of the digits of \(m + n\):
   \[
   m + n = 10^{100} + 98
   \]
   This is a 1 followed by 100 zeros, plus 98:
   \[
   10^{100} + 98 = 1000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000098
   \]

8. The sum of the digits of \(10^{100} + 98\) is:
   \[
   1 + 0 + 0 + \ldots + 0 + 9 + 8 = 1 + 9 + 8 = 18
   \]

The final answer is \(\boxed{18}\).