Consider some increasing, integer valued, function $f$ defined on the integers, such that $f(n)=\Theta(n\log n)$ when $n\to\infty$. For every $n$, let $g(n)=\inf\{k\mid f(k)\geqslant n\}$.
Now, for each $u_0$, define $(u_n)$ recursively by $u_{n+1}=g(u_n)$ for every $n$ and define $N(u_0)$ as the smallest $n$ such that $u_n\leqslant1$.
What are the asymptotics of $N(u_0)$ when $u_0\to\infty$?
My hope is that $N(n)=\Theta\left(\frac{\log n}{\log\log n}\right)$.
Short answer:
You can't answer, the asymptotic expression gives no clue about the initial values of $u$. $k$ could be $1$ and up to $\infty$.