Existence of a lower bound that goes to infinity used to be a long-standing problem of Rényi and Erdős. It was finally resolved by Schinzel with a bound of about $\log \log k$. The bound was later improved by Schinzel and Zannier to about $\log k$. Also, Zannier proved a lower bound on the number of terms of $g(f(x))$ for any $g$.

Given an example with $K(f^2)=A$ and $K(f)=B$, one can obtain a polynomial $g$ with $K(g^2)=A^2$ and $K(g)=B^2$. Indeed, consider $g(x)=f(x)f(x^{BIG})$ So, in view of the example posted by Richard Stanley in another answer, it follows that there is an $\varepsilon>0$ an infinitely many polynomials $f$ such that $K(f^2)\leq K(f)^{1-\varepsilon}$.

The question of whether the correct bound is logarithmic or polynomial in $k$ (or is in between) is open.

Existence of a lower bound that goes to infinity used to be a long-standing problem of Rényi and Erdős. It was finally resolved by Schinzel with a bound of about $\log \log k$. The bound was later improved by Schinzel and Zannier to about $\log k$. Also, Zannier proved a lower bound on the number of terms of $g(f(x))$ for any $g$.

Given an example with $K(f^2)=A$ and $K(f)=B$, one can obtain a polynomial $g$ with $K(g^2)=A^2$ and $K(g)=B^2$. Indeed, consider $g(x)=f(x)f(x^{BIG})$ So, in view of the example posted by Richard Stanley in another answer, it follows that there is an $\varepsilon>0$ and infinitely many polynomials $f$ such that $K(f^2)\leq K(f)^{1-\varepsilon}$.

The question of whether the correct bound is logarithmic or polynomial in $k$ (or is in between) is open.