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We investigate the presence of triangles in a class of correlated random graphs in which hidden variables determine the pairwise connections between vertices. The class rules out self-loops and multiple edges. We focus on the regime where the hidden variables follow a power law with exponent τ∈(2,3), so that the degrees have infinite variance. The natural cutoff hc characterizes the largest degrees in the hidden variable models, and a structural cutoff hs introduces negative degree correlations (disassortative mixing) due to the infinite-variance degrees. We show that local clustering decreases with the hidden variable (or degree). We also determine how the average clustering coefficient C scales with the network size N, as a function of hs and hc. For scale-free networks with exponent 2 <τ<3 and the default choices hs~N1/2 and hc~N1/(τ-1) this gives C~N2-τlnN for the universality class at hand. We characterize the extremely slow decay of C when τ≈2 and show that for τ=2.1, say, clustering starts to vanish only for networks as large as N=109. [ABSTRACT FROM AUTHOR]