List approximation for increasing Kolmogorov complexity

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. But is it possible to construct a few strings, not longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that on input a string $x$ of length $n$ returns a list with $O(n^2)$ many strings, all of length $n$, such that 99\% of them are more complex than $x$, provided the complexity of $x$ is less than $n - \log \log n - O(1)$. We obtain similar results for other parameters, including a polynomial-time construction.