We establish a scaling limit result for the length LIS\((\sigma_n)\) of the longest increasing subsequence of a permutation \(\sigma_n\) of size \(n\) sampled from the Brownian separable permuton \(\boldsymbol{\mu}_p\) of parameter \(p\in(0,1)\), which is the universal limit of pattern-avoiding permutations. Specifically, we prove that
\(\frac{\text{LIS}(\sigma_n)}{n^\alpha}\;\underset{n\to\infty}{\overset{\mathrm{a.s.}}{\longrightarrow}}\; X,\)
where \(\alpha=\alpha(p)\) is the unique solution in the interval \((1/2,1)\) to the equation
\(\frac{1}{4^{\frac{1}{2\alpha}}\sqrt{\pi}}\,\frac{\Gamma\big(\frac{1}{2}-\frac{1}{2\alpha}\big)}{\Gamma\big(1-\frac{1}{2\alpha}\big)}=\frac{p}{p-1},\)
and \(X=X(p)\) is a non-deterministic and a.s. positive and finite random variable, which is a measurable function of the Brownian separable permuton. Notably, the exponent \(\alpha(p)\) is an increasing continuous function of \(p\) with \(\alpha(0^+)=1/2,\) \(\alpha(1^-)=1\) and \(\alpha(1/2)\approx0.815226,\) which corresponds to the permuton limit of uniform separable permutations. We prove analogous results for the size of the largest clique of a graph sampled from the Brownian cographon of parameter \(p\in(0,1)\).