Reasonable estimate of an asymptotic limit notion of Kolmogorov complexity
As commenters @Lwins and @SashaNikolov have already stated, it is non-computably hard to compute a function as slow-growing as $f$. (So no really humanly comprehensible rate can be given.)

Namely, suppose we just have one direction of your equality: $f(c)\le\min_{x\ge c} K(x)$, but suppose $f$ does goes to infinity monotonically.

Then $x\ge c$ implies $K(x)\ge f(c)$. So given $n$, if we find $c$ with $f(c)\ge n$ then $K(c)\ge n$. So letting $F(n)=c$ we have a function $F$ with $K(F(n))\ge n$ for all $n$.

But according to Theorem 4.1 of my paper with Merkle and Stephan, this allows us to compute a diagonally non-computable function with finite range, i.e., it allows us to compute a complete extension of Peano arithmetic.