\textit{J. Aczél} showed in 1963 [see Math. Mag. 58, 42--45 (1985; Zbl 0571.39005)] that there is a simple functional equation involving two unknown functions, say \(f\) and \(g\), whose general solution (no regularity conditions whatever) is: \(f\) is a polynomial of degree at most 2 and \(g\) is the derivative of \(f\). The authors extend Aczél's result by showing that there is a sequence \(E_n\) of functional equations each involving functions \(f\) and \(g\), such that for any \(n\geq 1\), the general solution of \(E_n\), again without any regularrity conditions whatever, is: \(f\) is a polynomial of degree at most \(2n\) and \(g\) is the derivative of \(f\).