Surprising appearances of Painlevé transcendents What are some of your favorite examples of enumerative problems whose answer ended up being (related to) a solution to one of the Painlevé equations?
I have seen examples from enumeration of classes of graphs, to Hodge integrals and Hurwitz numbers and I'm sure there are many more out there. Bonus points if there is an explanation/heuristic that demystifies or explains why one should expect coefficients of a Painlevé transcendent to appear.
 A: The OP likely has surprising appearances in the mathematical literature in mind, but perhaps the physical observation of a Painléve transcendent is noteworthy?   
In Universal Fluctuations of Growing Interfaces: Evidence in Turbulent Liquid Crystals (2001) the growth of a liquid crystal cluster is observed in time. The time-dependent probability distribution $\rho$ of the radius $R$ follows (upon rescaling) the solution of the Painléve II equation (dashed curve in the right plot). The origin of the correspondence is described by Craig Tracy in Integrable Probability and the Role of Painlevé Functions.
I am not aware of any other real-world observation of a Painlevé function.

A: I have seen Painlevé transcendents appear a few times in the context of field theory.  For example, they appear in model solutions of Hitchin's equations for Higgs fields.  (See [MSWW16] and references therein.)
I have also discovered the Hastings-McLeod solution in a certain rescaling limit of the Ginzburg-Landau equations.  (Unfortunately this is unpublished.)
Both instances occurred in the context of radially-equivariant solutions to a non-linear elliptic PDE in two dimensions.  The resulting ODE is unstable, and there is a distinguished solution which is well-behaved at both $r=0$ and $r=\infty$.
If you have a reasonably-natural nonlinear ODE which is not Painléve, you might be able to add some parameters, take a limit, get lucky, and find something which is Painléve.  I also have a vague intuition that their appearance is related to their nice asymptotics.  
[MSWW16] Mazzeo, Rafe; Swoboda, Jan; Weiss, Hartmut; Witt, Frederik, Ends of the moduli space of Higgs bundles, Duke Math. J. 165, No. 12, 2227-2271 (2016). ZBL1352.53018. 
