This talk concerns singular values of M-fold products of i.i.d. right-unitarily invariant N x N random matrix ensembles. As N tends to infinity, the height function of the Lyapunov exponents converges to a deterministic limit by work of Voiculescu and Nica-Speicher for M fixed and by work of Newman and Isopi-Newman for M tending to infinity with N. In this talk, I will show for a variety of ensembles that fluctuations of these height functions about their mean converge to explicit Gaussian fields which are log-correlated for M fixed and have a white noise component for M tending to infinity with N. These ensembles include rectangular Ginibre matrices, truncated Haar-random unitary matrices, and right-unitarily invariant matrices with fixed singular values. I will sketch our technique, which derives a central limit theorem for global fluctuations via certain conditions on the multivariate Bessel generating function, a Laplace-transform-like object associated to the spectral measures of these matrix products. This is joint work with Vadim Gorin.