Natural genetic circuits enable cells to make sophisticated digital decisions. Building equally complex synthetic circuits in eukaryotes remains difficult, however, because commonly used components leak transcriptionally, do not arbitrarily interconnect or do not have digital responses. Here, we designed dCas9-Mxi1-based NOR gates in Saccharomyces cerevisiae that allow arbitrary connectivity and large genetic circuits. Because we used the chromatin remodeller Mxi1, our gates showed minimal leak and digital responses. We built a combinatorial library of NOR gates that directly convert guide RNA (gRNA) inputs into gRNA outputs, enabling the gates to be 'wired' together. We constructed logic circuits with up to seven gRNAs, including repression cascades with up to seven layers. Modelling predicted the NOR gates have effectively zero transcriptional leak explaining the limited signal degradation in the circuits. Our approach enabled the largest, eukaryotic gene circuits to date and will form the basis for large, synthetic, cellular decision-making systems.