For optimality results in sequential analysis which contain explicit description of optimal strategies the assumption of linear costs of observation ct has been of major importance. In this paper we leave this assumption and investigate the question how the shape of nonlinear costs of observation influences the shape of optimal stopping boundaries. We do this for an optimal stopping problem which arises in the derivation of locally most powerful sequential tests for the Wiener process. It is found that costs of observation of the formlead to optimal stopping boundaries which grow in the order ofas t → ∞