In medical imaging, high-resolution can be crucial for identifying pathologies and subtle changes in tissue structure. However, in many scenarios, achieving high image resolution can be limited by physics or available technology. In this paper, we aim to develop an automatic and fast approach to increasing the resolution of Optical Coherence Tomography (OCT) images using the data available, without any additional information or repeated scans. We adapt a fully connected deep learning network for the super-resolution task, allowing multi-scale similarity to be considered, and create a training and testing set of more than 40,000 sample patches from retinal OCT data. Testing our model, we achieve an impressive root mean squared error of 5.847 and peak signal-to-noise ratio (PSNR) of 33.28 dB averaged over 8282 samples. This represents a mean improvement in PSNR of 3.2 dB over nearest neighbour and 1.4 dB over bilinear interpolation. The results achieved so far improve over commonly used fast techniques for increasing resolution and are very encouraging for further development towards fast OCT super-resolution. The ability to increase quickly the resolution of OCT as well as other medical images has the potential to impact significantly on medical imaging at point of care, allowing significant small details to be revealed efficiently and accurately for inspection by clinicians and graders and facilitating earlier and more accurate diagnosis of disease.