Growth curves of the ESR dating signal at g = 2.0006 on irradiation have been carefully examined. They revealed that a description by a single exponential saturation function is only a good approximation in the dose range smaller than 1 kGy. In this dose range the saturation dose, D0, for carbonates was found to be between 400 and 800 Gy. The systematic deviations from this function and statistical uncertainties limit the reliable AD range to about 250 Gy. Expanded functions and models that explain the observed deviations are given. The usual way of determining the thermal life time of ESR signals (isothermal annealing and Arrhenius plot) is critically discussed and additional specific complications (trap decay, different components of the signal) of the signal at g = 2.0006 are shown. A different method of detecting long term fading was used, in which the natural signal growth is compared with artificial growth curves on gamma-irradiation. Evaluation of measurements from the sediment core RC17-177 revealed a thermal stability lower than previously assumed for foraminifera. The data fit best with the assumption that the dating signal contains a component (about 50%) that has a very low thermal stability (with a lifetime of about 50 ka at 2-degrees-C). Annealing experiments indicate that this may be the case in other carbonates as well. This would drastically limit the dating applicability of the signal at g = 2.0006.