Graph Neural Networks (GNNs) have achieved unprecedented success in identifying categorical labels of graphs. However, most existing graph classification problems with GNNs follow the protocol of balanced data splitting, which misaligns with many real-world scenarios in which some classes have much fewer labels than others. Directly training GNNs under this imbalanced scenario may lead to uninformative representations of graphs in minority classes, and compromise the overall classification performance, which signifies the importance of developing effective GNNs towards handling imbalanced graph classification. Existing methods are either tailored for non-graph structured data or designed specifically for imbalanced node classification while few focus on imbalanced graph classification. To this end, we introduce a novel framework, Graph-of-Graph Neural Networks (G$^2$GNN), which alleviates the graph imbalance issue by deriving extra supervision globally from neighboring graphs and locally from stochastic augmentations of graphs. Globally, we construct a graph of graphs (GoG) based on kernel similarity and perform GoG propagation to aggregate neighboring graph representations. Locally, we employ topological augmentation via masking node features or dropping edges with self-consistency regularization to generate stochastic augmentations of each graph that improve the model generalibility. Extensive graph classification experiments conducted on seven benchmark datasets demonstrate our proposed G$^2$GNN outperforms numerous baselines by roughly 5\% in both F1-macro and F1-micro scores. The implementation of G$^2$GNN is available at https://github.com/YuWVandy/G2GNN}{https://github.com/YuWVandy/G2GNN