  We consider powers of the absolute value of the characteristic polynomial of
Haar distributed random orthogonal or symplectic matrices, as well as powers of
the exponential of its argument, as a random measure on the unit circle minus
small neighborhoods around $\pm 1$. We show that for small enough powers and
under suitable normalization, as the matrix size goes to infinity, these random
measures converge in distribution to a Gaussian multiplicative chaos measure.
Our result is analogous to one on unitary matrices previously established by
Christian Webb in [31]. We thus complete the connection between the classical
compact groups and Gaussian multiplicative chaos. To prove this we establish
appropriate asymptotic formulae for Toeplitz and Toeplitz+Hankel determinants
with merging singularities. Using a recent formula communicated to us by Claeys
et al., we are able to extend our result to the whole of the unit circle.
