Graphons are analytic objects representing limits of convergent sequences of graphs. Lov\'asz and Szegedy conjectured that every finitely forcible graphon, i.e. any graphon determined by finitely many subgraph densities, has a simple structure. In particular, one of their conjectures would imply that every finitely forcible graphon has a weak \(\varepsilon\)-regular partition with the number of parts bounded by a polynomial in \(\varepsilon^{-1}\). We construct a finitely forcible graphon \(W\) such that the number of parts in any weak \(\varepsilon\)-regular partition of \(W\) is at least exponential in \(\varepsilon^{-2}/2^{5\log^*\varepsilon^{-2}}\). This bound almost matches the known upper bound for graphs and, in a certain sense, is the best possible for graphons.