We present a new adaptive higher-order finite element method (hp-FEM) for the solution of boundary value problems formulated in terms of partial differential equations (PDEs). The method does not use any information about the problem to be solved which makes it robust and equation-independent. It employs a higher-order reconstruction scheme over local element patches which makes it faster and easier to parallelize compared to hp-adaptive methods that are based on the solution of a reference problem on a globally hp-refined mesh. The method can be used for the solution of linear as well as nonlinear problems discretized by conforming or non-conforming finite element methods, and it can be combined with arbitrary a posteriori error estimators. The performance of the method is demonstrated by several examples carried out by the discontinuous Galerkin method.