Summary: Let \(\mathbb{F}\) be a commutative ring. A restricted skew polynomial extension over \(\mathbb{F}\) is a class of iterated skew polynomial \(\mathbb{F}\)-algebras which include well-known quantized algebras such as the quantum algebra \(U_q (\mathfrak{sl}_2)\), Weyl algebra, etc. Here we obtain a necessary and sufficient condition in order to be restricted skew polynomial extensions over \(\mathbb{F}\). We also introduce a restricted Poisson polynomial extension which is a class of iterated Poisson polynomial algebras and observe that a restricted Poisson polynomial extension appears as semiclassical limits of restricted skew polynomial extensions. Moreover, we obtain usual as well as unusual quantized algebras of the same Poisson algebra as applications.