Recent advances in long-read sequencing (LRS) and assembly algorithms have made it possible to create highly complete genome assemblies for humans, animals, plants, and other eukaryotes. However, there is a need for ongoing development to improve accessibility and affordability of the required data, increase the range of usable sample types, and reliably resolve the most challenging, repetitive genome regions. 'Cornetto' is a new experimental paradigm in which the genome assembly process is adaptively integrated with programmable selective nanopore sequencing, with target regions being iteratively updated to focus LRS data production onto the unsolved regions of a nascent assembly. This improves assembly quality and streamlines the process, both for human individuals and diverse non-human vertebrates, including endemic Australian endangered species, tested here. Cornetto enables us to generate highly complete diploid human genome assemblies using only a single LRS platform, surpassing the quality of previous efforts at a fraction of the cost. Cornetto enables genome assembly from challenging sample types like human saliva, for the first time, further enhancing accessibility. Finally, we obtain complete and accurate assemblies for clinically-relevant repetitive loci at the extremes of the genome, demonstrating valid approaches for genetic diagnosis in facioscapulohumeral muscular dystrophy (FSHD) and MUC1-autosomal dominant tubulointerstitial kidney disease (MUC1-ADTKD) - inherited diseases for which diagnosis is complicated by an inability to sequence the genes involved. In summary, Cornetto will improve, accelerate, and democratise genome assembly, delivering impacts across a range of bioscience domains.