In the theory of dense graph limits, a graphon is a symmetric measurable function $W:[0,1]^2\to [0,1]$. Each graphon gives rise naturally to a random graph distribution, denoted $\mathbb{G}(n,W)$, that can be viewed as a generalization of the Erdős-Rényi random graph. Recently, Doležal, Hladký, and Máthé gave an asymptotic formula of order $\log n$ for the clique number of $\mathbb{G}(n,W)$ when $W$ is bounded away from 0 and 1. We show that if $W$ is allowed to approach 1 at a finite number of points, and displays a moderate rate of growth near these points, then the clique number of $\mathbb{G}(n,W)$ will be $Θ(\sqrt{n})$ almost surely. We also give a family of examples with clique number $Θ(n^α)$ for any $α\in(0,1)$, and some conditions under which the clique number of $\mathbb{G}(n,W)$ will be $o(\sqrt{n})$, $ω(\sqrt{n}),$ or $Ω(n^α)$ for $α\in(0,1)$.