Modern microscopes can image at atomic resolutions but often reach technical limitations for high-resolution images captured at the smallest nanoscale. Prior works have applied super-resolution (SR) by deep neural networks employing high-resolution images as targets in supervised training. However, in practice, it may be impossible to obtain these high-resolution images at the smallest atomic scales. Approaching this problem, we consider a new super-resolution training paradigm based on low-resolution (LR) microscope images only, to surpass the highest physically captured resolution available for training. As a solution, we propose a novel multi-scale training method for SR based on LR data only, which simultaneously supervises SR at multiple resolutions, allowing the SR to generalize beyond the LR training data. We physically captured low- and high-resolution images for evaluation, thereby incorporating real microscope degradation to deliver a proof of concept. Our experiments on periodic atomic structure in STEM and STM microscopy images show that our proposed multi-scale training method enables deep neural network image SR even up to 360% of the highest physically recorded resolution. Code and data is available on github.