What's theoretical maximum information compression rate? Let's say I've got a random bit sequence s and a reversible function f(s), for which the following statement f'(f(s)) = s is true. What is the theoretical maximum average compression rate of such function?
IIRC, most if not all compression algorithms of today tend to identify particular patterns and map them with standard where it's possible. This makes the maximum and minimum compression rate 1/[s] and 1 correspondingly.
NB: Higher rate — weaker compression (worse)

The optimum average length of the compressed binary stream $f(s^{[n]})$, (let's call it $L^{[n]}$) is given by the first Shannon theorem, in the range:
$$ H(s^{[n]})\le L^{[n]}<H(s^{[n]})+1$$
so the compression rate $r=L^{[n]}/n$ ($n=[s]$ in your notation) is 
$$ \frac{H(s^{[n]})}{n}\le r<\frac{H(s^{[n]})}{n}+\frac{1}{n}$$
Here $H(s^{[n]})$ is the entropy (in bits) of the full message of length $n$. In the particular case of independent stationary symbols, $H(s^{[n]})=n\, H(s)$ where $H(s)$ is the entropy of a single bit, and then
$$ H(s)\le r< H(s) +\frac{1}{n}$$
