If the Collatz conjecture is strongly false, in the sense that there is an infinite orbit, let $S_n$ be the set of natural numbers $\le n$ whose orbit goes to infinity.
If $c=\liminf _{n\rightarrow\infty} |S_n|/\log_2(n)$ it's easy to see that $c=\infty$.
Could one prove some stronger results, such as, for example: $\liminf_{n\rightarrow\infty} \log(|S_n|)/\log\log(n)>1$?
Krasikov and Lagarias prove the following result (Theorem 6.1):
For any $a\neq 0\pmod 3$, and for all sufficiently large $x$ (depending on $a$), there are at least $x^{0.84}$ integers below $x$ such that iteration of Collatz map will at some point reach $a$.
If $a$ is a counterexample to the Collatz conjecture, then so is any number which eventually reaches $a$, and they are counterexamples of the same type - trajectory tends to infinity if that of $a$ does, and else it falls into the same loop as $a$. Thus if we assume that there is at least one number whose trajectory tends to infinity, we get $\liminf \log|S_n|/\log n\geq 0.84$, much stronger than what you've asked.
If we merely assume Collatz conjecture is false, this is not enough to prove $S_n$ is nonempty, but if you expand $S_n$ to include numbers falling into nontrivial loops, we get the same bound. Though those papers don't treat your exact question, I suspect this is still state of the art for lower bounds.
As for upper bounds, I don't think we can give anything better than $|S_n|\leq n-n^{0.84}$ which also follows from that theorem.
