Products of random matrices have practical applications in many fields of science. If the product consists of independent Gaussian random matrices with independent, identically distributed centered real entries, then such matrices are usually called real Ginibre matrices. In this special case, one can analytically derive the whole spectrum of Lyapunov exponents. Furthermore, the calculation uncovers a connection between the spectrum and the law of large numbers. In the paper, the authors show that the distribution of each Lyapunov exponent is normal for a large number of product matrices. Furthermore, they rederive Newman’s triangular law which has an interpretation as the radial density of complex eigenvalues in the circular law and study the commutativity of the two limits, when the number of matrices in the product tends to infinity as well as the dimension of matrices in the product, on the global and the local scale. Finally, it is shown that an asymptotic expansion of a Meijer G-function with large index leads to a Gaussian.