We consider the problem of sequential parameter estimation of a nonlinear function under the Bayesian setting. The designer can choose inputs for a sequence of experiments to obtain an accurate estimate of the system parameters based on observed outputs, while complying with a constraint on the expected outputs of the system. We quantify the accuracy of the obtained estimate in terms of the pound 2 norm. We propose to solve the problem by casting it as the problem of minimizing the Bayesian Mean Square Error (BMSE) of the parameter estimate subject to a constraint on the expected deviation of the output from the desired target value. We develop a greedy policy to solve the problem in the sequential setting, and we characterize the solution structure based on analytical results for the Gaussian case. For a computationally tractable update of the posterior, we propose the use of a surrogate model combined with approximate Bayesian computation. We evaluate the proposed approach on the use case of smart road compaction, where the goal is to estimate asphalt parameters while reaching the desired compaction level, by choosing the value of the loading pressure. Simulation results on a synthetic road compaction dataset show the efficacy of the proposed solution scheme in both parameter estimation and effective compaction of the road.