  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 graph 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.
