Physical and functional constraints on biological networks lead to complex topological patterns across multiple scales in their organization. A particular type of higher-order network feature that has received considerable interest is network motifs, defined as statistically regular subgraphs. These may implement fundamental logical and computational circuits and are referred as "building blocks of complex networks". Their well-defined structures and small sizes also enables the testing of their functions in synthetic and natural biological experiments. The statistical inference of network motifs is however fraught with difficulties, from defining and sampling the right null model to accounting for the large number of possible motifs and their potential correlations in statistical testing. Here we develop a framework for motif mining based on lossless network compression using subgraph contractions. The minimum description length principle allows us to select the most significant set of motifs as well as other prominent network features in terms of their combined compression of the network. The approach inherently accounts for multiple testing and correlations between subgraphs and does not rely on a priori specification of an appropriate null model. This provides an alternative definition of motif significance which guarantees more robust statistical inference. Our approach overcomes the common problems in classic testing-based motif analysis. We apply our methodology to perform comparative connectomics by evaluating the compressibility and the circuit motifs of a range of synaptic-resolution neural connectomes.