It's been a while since I had to deal with some sort of asymptotic analysis so I am a bit rusty and trying to get the basics back together. Since I don't really know where to look for these things, I ask you: is the following correct?
Let $p_i,q_i \in \mathbb R$ with $0<p_i<q_i$. Let $a_i, b_i\in \mathbb R$ with $1<a_i<b_i$. Then we have
$ \log(n) \ll n^{p_1} \ll n^{q_1} \ll a_2^{n^{p_2}} \ll b_2^{n^{p_2}} \ll a_3^{n^{{q_2}}} \ll n! \ll n^n$
And: would you add anything to that list? (Take this as a "soft question". I think it might be possible, with suitable products with logarithms, to add an infinite sequence at every step, but I mean "atomic sequences", whatever that may mean.)
Let me clarify on the use of the $\ll$ symbol. I write $a_n\ll b_n$ if $\lim_n \frac{b_n}{a_n}=\infty$.