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Although population-level analyses revealed significant roles for CTCF and cohesin in mammalian genome organization, their contributions at the single-cell level remain incompletely understood. Here, we used a super-resolution microscopy approach to measure the effects of removal of CTCF or cohesin in mouse embryonic stem cells. Single-chromosome traces revealed cohesin-dependent loops, frequently stacked at their loop anchors forming multi-way contacts (hubs), bridging across TAD boundaries. Despite these bridging interactions, chromatin in intervening TADs was not intermixed, remaining separated in distinct loops around the hub. At the multi-TAD scale, steric effects from loop stacking insulated local chromatin from ultra-long range (>4 Mb) contacts. Upon cohesin removal, the chromosomes were more disordered and increased cell-cell variability in gene expression. Our data revise the TAD-centric understanding of CTCF and cohesin and provide a multi-scale, structural picture of how they organize the genome on the single-cell level through distinct contributions to loop stacking. [Display omitted] • Chromosome tracing shows radially organized chromosome loops • CTCF boundary sites form hubs that depend on cohesin and CTCF • Loss of cohesin leads to expansion at the <4 Mb scale and increased mixing at >4 Mb scale • Loss of cohesin increases entropy in genome folding and variation in gene expression Hafner et al. use microscopy to trace folding of individual chromosomes. They observe and quantify chromatin loops and find that loop stacking is important for 3D organization across genomic scales. CTCF is important for preferential clustering of TAD-boundary sites. Cohesin is essential for both TAD and chromosome-scale folding. [ABSTRACT FROM AUTHOR]