Single domain generalization aims to train a generalizable model with only one source domain to perform well on arbitrary unseen target domains. Image augmentation based on Random Convolutions (RandConv), consisting of one convolution layer randomly initialized for each mini-batch, enables the model to learn generalizable visual representations by distorting local textures despite its simple and lightweight structure. However, RandConv has structural limitations in that the generated image easily loses semantics as the kernel size increases, and lacks the inherent diversity of a single convolution operation. To solve the problem, we propose a Progressive Random Convolution (Pro-RandConv) method that recursively stacks random convolution layers with a small kernel size instead of increasing the kernel size. This progressive approach can not only mitigate semantic distortions by reducing the influence of pixels away from the center in the theoretical receptive field, but also create more effective virtual domains by gradually increasing the style diversity. In addition, we develop a basic random convolution layer into a random convolution block including deformable offsets and affine transformation to support texture and contrast diversification, both of which are also randomly initialized. Without complex generators or adversarial learning, we demonstrate that our simple yet effective augmentation strategy outperforms state-of-the-art methods on single domain generalization benchmarks.