Serum stimulation of a number of different mouse cell lines results in sustained oscillations of Hes1, a member of this Hes/Her family of transcription factors. Quantitative time-course expression data obtained in this system provide an excellent opportunity to explore transcriptional oscillations in a relatively simple setting. Simple models of the Hes1 regulatory circuit are capable of generating oscillations that share many features with those observed in mouse fibroblasts, and highlight the central role played by delayed negative feedback. However, taking into account constraints on model parameters imposed by experimental data, these models can only generate oscillations with quite low peak-to-trough expression ratios. To explore the origin of this limitation, we develop a more detailed model of the Hes1 circuit, incorporating nucleo-cytoplasmic transport, Hes1 dimerisation, and differential stability of Hes1 monomers and dimers. We show that differential protein stability can increase the amplitude of Hes1 oscillations, but that the resulting expression profiles do not fully match experimental data. We extend the model by incorporating periodic forcing of the Hes1 circuit by cyclic phosphorylation of the protein Stat3. We show that time delays and differential stability act synergistically in this extended model to generate large amplitude oscillatory solutions that match the experimental data well.