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We study the 'critical moments' of subcritical Gaussian multiplicative chaos(GMCs) in dimensions $d \leq 2$. In particular, we establish a fully explicitformula for the leading order asymptotics, which is closely related to largedeviation results for GMCs and demonstrates a similar universality feature. Weconjecture that our result correctly describes the behaviour of analogousmoments of moments of random matrices, or more generally structures which areasymptotically Gaussian and log-correlated in the entire mesoscopic scale. Thisis verified for an integer case in the setting of circular unitary ensemble,extending and strengthening the results of Claeys et al. and Fahs tohigher-order moments.

On the critical-subcritical moments of moments of random characteristic polynomials: a GMC perspective