  Using asymptotics of Toeplitz+Hankel determinants, we establish formulae for
the asymptotics of the moments of the moments of the characteristic polynomials
of random orthogonal and symplectic matrices, as the matrix-size tends to
infinity. Our results are analogous to those that Fahs obtained for random
unitary matrices in [14]. A key feature of the formulae we derive is that the
phase transitions in the moments of moments are seen to depend on the symmetry
group in question in a significant way.
