We give a brief review on non-formal star products and star exponentials and star functions (Omori et al., Deformation of expressions for elements of an algebra, in Symplectic, Poisson, and Noncommutative Geometry. Mathematical Sciences Research Institute Publications, vol. 62 (Cambridge University Press, Cambridge, 2014), pp. 171–209; Deformation of expressions for elements of algebra, arXiv:1104.1708v1[math.ph]; Deformation of expressions for elements of algebras (II), arXiv:1105.1218v2[math.ph]). We introduce a star product on polynomials with a deformation parameter ħ > 0. Extending to functions on complex space enables us to consider exponential element in the star product algebra, called a star exponential. By means of the star exponentials we can define several functions called star functions in the algebra, with some noncommutative identities. We show certain examples.