We undertake an asymptotic study of a second Painlevé hierarchy based on the Jimbo-Miwa Lax pair in the limit as the independent variable approaches infinity. The hierarchy is defined by an infinite sequence of non-linear ordinary differential equations, indexed by order, with the classical second Painlevé equation as the first member. We investigate general and special asymptotic behaviours admitted by each equation in the hierarchy. We show that the general asymptotic behaviour is described by two related hyperelliptic functions, where the genus of the functions increases with each member of the hierarchy, and we prove that there exist special families of solutions which are represented by algebraic formal power series. For specific values of the constants which appear in the higher-order second Painlevé equations, exact solutions are also constructed. Particular attention is given to the fourth-order analogue of the classical second Painlevé equation. In this case, the general asymptotic behaviour is given to leading-order by two related genus-2 hyperelliptic functions. These functions are characterised by four complex parameters which depend on the independent variable through the perturbation terms of the leading-order equations, and we investigate how these parameters change with respect to this variable. We also show that the fourth-order equation admits two classes of algebraic formal power series and that there exist families of true solutions with these behaviours in specified sectors of the complex plane, as well as unique solutions in extended sectors. To complement our asymptotic study of higher-order Painlevé equations, we consider a new setting in which classical Painlevé equations arise. We study reaction-diffusion equations with quadratic and cubic source terms, with a spatio-temporal dependence included in those terms, and show that solutions of these equations are given by first and second Painlevé transcendents.