Let (X,B,P) be a standard probability space. Let T:C\rightarrow PPT(X) be a free action of the complex plane on the space (X,B,P). We say that the function F:X\rightarrow C is measurably entire if it is measurable and for P-a.e x the function F_x(z):=F(T_zx) is entire. B. Weiss showed in '97 that for every free C action there exists a non-constant measurably entire function. In the talk I will present upper and lower bounds for the growth of such functions. The talk is partly based on a joint work with L. Buhovsky, A.Logunov, and M. Sodin.