Traditional hyperspectral (HS) pansharpening aims at fusing a HS image with its panchromatic (PAN) counterpart, to bring the spatial resolution of the HS image to that of the PAN image. However, in many practical applications, arbitrary resolution HS (ARHS) pansharpening is required, where the HS and PAN images need to be integrated to generate a pansharpened HS image with arbitrary resolution (usually higher than that of the PAN image). Such an innovative task brings forth new challenges for the pansharpening technique, mainly including how to reconstruct HS images beyond the training scale and how to guarantee spectral fidelity at any spatial resolutions. To tackle the challenges, we present a novel convolutional neural network (CNN)-based method for ARHS pansharpening called ARHS-CNN. It is based on a two-step relay optimization process, which is associated with a multilevel enhancement subnetwork and a rescaling subnetwork. With a careful design following the thread, our ARHS-CNN is able to pansharpen HS images to any spatial resolutions using just a single CNN model trained on a limited number of scales while meantime to keep spectral fidelity at those resolutions, which wins an obvious advantage over traditional pansharpening methods. Experimental results obtained on several datasets verify the excellent performance of our ARHS-CNN method.