Is Kolmogorov complexity quasi-surjective?
Suppose $L_0$ is an essentially-optimal description language, and consider the identity function, which is a computable partial function from $\{0,1\}^*$ to itself. According to the definition, there is a computable function $f$ and a constant $C$ such that $|f(x)| \leq |x| + C$ and $L_0(f(x)) = x$.

Take now a Kolmogorov random string $x$ of length $n$. On the one hand, $K(x) \geq n$. On the other hand, we have seen above that $K(x) \leq n + C$.