In this paper, a new class of very-high-order finite difference TENO schemes with discontinuity-resolving property for compressible Euler equations is proposed. To achieve arbitrary high-order schemes within a unified framework, a novel dual ENO-like stencil selection strategy is proposed based on the candidate stencil arrangement with incremental width. Based on a tailored coupling strategy, the first three small candidate stencils not only provide the reference to judge whether the candidate stencil with a large width is smooth or not but also determine whether the targeted cell is being crossed by genuine discontinuities or not, by combining the stencil smoothness information of several adjacent cells. Without losing generality, the typical seventh- and ninth-order TENO-DR schemes are constructed in detail and a variety of benchmark-test problems, including broadband waves, strong shock and contact discontinuities are studied. Compared to the state-of-the-art TENO schemes, the present schemes exhibit significantly improved high-resolution property and sharp shock-capturing capability, and thus are promising for more complex compressible flow simulations where the excessive numerical dissipation of existing schemes prevents the concerned flow structures from being properly resolved.