Viscosifiers, especially terminated polymers that have functional terminal groups, the polymers being pre-extended by polyols and being reacted to give polymers that are terminated by other functional groups, are produced. The viscosifiers have a low content in educts or educt descendants that deteriorate the properties of compositions. The formation of high-molecular addition products is considerably reduced or even excluded so that the products obtained have low viscosity and good storage stability. The epoxy-terminated polymer of formula (I) is especially preferred.