We devise and study experimentally adaptive strategies driven by a posteriori error estimates to select automatically both the space mesh and the polynomial degree in a numerical approximation of the Laplace equation in two space dimensions. The adaptation is based on equilibrated flux estimates. These estimates are presented here for inhomogeneous Dirichlet and Neumann boundary conditions and for spatially-varying polynomial degree; they deliver a global error upper bound with constant one and, up to data oscillation, error lower bounds on element patches with a (computable) generic constant independent of the mesh size and of the polynomial degree. We numerically asses the estimates and the hp-adaptive strategy using the interior penalty discontinuous Galerkin method. Asymptotic exactness is observed for all the symmetric, nonsymmetric, and incomplete variants on non-nested unstructured grids for a smooth solution. Exponential convergence rates are reported for the incomplete version on several benchmarks with a singular solution.