The fractal dimensions of fractal sets generated by iterated function systems (IFS) have been extensively studied. Among their fundamental properties, the most classical and basic one is Bowen's formula, which allows us to determine the Hausdorff dimension of such fractal sets by considering only the natural coverings generated by the IFS. Since the 1980s, random fractal sets generated by random iterated function systems have been introduced, and under various assumptions, a version of Bowen's formula have been established. However, there has been no study of random iterated function systems for which the random version of Bowen's formula fails to hold. In this talk, I will present a construction of random iterated function systems for which the random version of Bowen's formula fails to hold.