When does $1.00001^n$ exceed $n^{100001}$?
The $n$ where the two expressions are equal isn't all that huge. Here's how to calculate it.

We want
$$ 1.000001^n \approx n^{1000001}.$$
This is the same as
$$e^{n \log 1.000001} \approx e^{1000001 \log n}.$$
Now, taking the $\log$ of both sides, and using Taylor's formula to approximate the left-hand side, we get
$$ n*10^{-6} \approx 10^6 \log n,$$
or
$$\frac{n}{\log n} \approx 10^{12}.$$
Since this means $\log n \approx \log 10^{12},$ we get $$n \approx 10^{12} \log 10^{12} = 2.7 * 10^{13}. $$
More exactly, using Maple to solve the equation numerically (you have to give it the estimate above as a starting value or it gives up), we find that $n \approx 3.1*10^{13}$.