I am looking for a list of all the bounds on $c_n$, the $n$-th composite. There is a trivial bound $2n \geq c_n >n$ $\forall n \geq 5$. But I am looking for bounds stronger than this. I have searched in Wikipedia but the information there didn't satisfy me.
Any help will be appreciated. Also it will be better if it would be moentioned that which among the bounds can be proven using elementary methods (the word 'elementary' being used in usual sense).
$c_n - n - 1 = \pi(c_n)$, so you can use bounds on the prime counting function.
From $n \leq c_n$, we can derive another simple one by using one of these:
$$ c_n - n - 1 = \pi(c_n) < \frac{c_n}{\ln c_n - 4} \leq \frac{c_n}{\ln n - 4}$$
and then solve for $c_n$ to get an upper bound. (and similarly to get a lower bound)