Time-bounded Kolmogorov complexity and non-random strings
I suspect that there is something that can be said generally, but off the cuff, the best I can see is that as $t$ increases, the number of compressible strings increases as well.

On the other hand, we certainly cannot pick a $t$ such that $f$ is polynomial.  Notice that if $t(|x|)&lt;|x|$, then there isn't enough time to write anything.  On the other hand, strings of the form $yyz$ can be generated by programs of length $|y|+|z|&lt;|x|$ and there are at least $2^{|x|/2}$ of those.  So by monotonicity, $f$ will always be at least this large.