This paper presents a novel nonmyopic adaptive Gaussian process planning (GPP) framework endowed with a general class of Lipschitz continuous reward functions that can unify some active learning/sensing and Bayesian optimization criteria and offer practitioners some flexibility to specify their desired choices for defining new tasks/problems. In particular, it utilizes a principled Bayesian sequential decision problem framework for jointly and naturally optimizing the exploration-exploitation trade-off. In general, the resulting induced GPP policy cannot be derived exactly due to an uncountable set of candidate observations. A key contribution of our work here thus lies in exploiting the Lipschitz continuity of the reward functions to solve for a nonmyopic adaptive $\epsilon$-optimal GPP ($\epsilon$-GPP) policy. To plan in real time, we further propose an asymptotically optimal, branch-and-bound anytime variant of $\epsilon$-GPP with performance guarantee. We empirically demonstrate the effectiveness of our $\epsilon$-GPP policy and its anytime variant in Bayesian optimization and a new task using both synthetic and real-world datasets.