Estimating arbitrary quantities of interest (QoIs) that are non-linear operators of complex, expensive-to-evaluate, black-box functions is a challenging problem due to missing domain knowledge and finite budgets. Bayesian optimal design of experiments (BODE) is a family of methods that identify an optimal design of experiments (DOE) under different contexts, using only in a limited number of function evaluations. Under BODE methods, sequential design of experiments (SDOE) accomplishes this task by selecting an optimal sequence of experiments while using data-driven probabilistic surrogate models instead of the expensive black-box function. Probabilistic predictions from the surrogate model are used to define an information acquisition function (IAF) which quantifies the marginal value contributed or the expected information gained by a hypothetical experiment. The next experiment is selected by maximizing the IAF. A generally applicable IAF is the expected information gain (EIG) about a QoI as captured by the expectation of the Kullback-Leibler divergence between the predictive distribution of the QoI after doing a hypothetical experiment and the current predictive distribution about the same QoI. We model the underlying information source as a fully-Bayesian, non-stationary Gaussian process (FBNSGP), and derive an approximation of the information gain of a hypothetical experiment about an arbitrary QoI conditional on the hyper-parameters The EIG about the same QoI is estimated by sample averages to integrate over the posterior of the hyper-parameters and the potential experimental outcomes. We demonstrate the performance of our method in four numerical examples and a practical engineering problem of steel wire manufacturing. The method is compared to two classic SDOE methods: random sampling and uncertainty sampling.