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Robust optimization is a popular paradigm for modeling and solving two- and multi-stage decision-making problems affected by uncertainty. In many real-world applications, the time of information discovery is decision-dependent and the uncertain parameters only become observable after an often costly investment. Yet, most of the literature assumes that uncertain parameters can be observed for free and that the sequence in which they are revealed is independent of the decision-maker's actions. To fill this gap. we consider two- and multi-stage robust optimization problems in which part of the decision variables control the time of information discovery. Thus, information can be discovered (at least in part) by making strategic exploratory investments in previous stages. We propose a novel dynamic formulation of the problem and prove its correctness. We leverage our model to provide a solution method inspired from the K-adaptability approximation. We reformulate the problem as a finite mixed-integer (resp. bilinear) program if none (resp. some of the) decision variables are real-valued. This finite program is solvable with off-the-shelf solvers. We generalize our approach to the minimization of piecewise linear convex functions. We demonstrate the effectiveness of our method in terms of interpretability, optimality, and speed on synthetic instances of the Pandora box problem, the preference elicitation problem, the best box problem, and the R&D project portfolio optimization problem. Finally, we evaluate it on an instance of the active preference elicitation problem used to recommend kidney allocation policies to policy-makers at the United Network for Organ Sharing.