Rainfall forecasting presents a dual challenge: extreme \emph{zero inflation}, where dry days dominate and obscure meaningful precipitation patterns, and pronounced \emph{nonstationarity}, where climate dynamics evolve across time and regimes. We propose the \textbf{Deep Extreme Transformer (DET)}, a principled architecture that integrates statistical distribution modeling with neural sequence learning to address both issues simultaneously. DET augments the Transformer with a Tweedie distribution output head that unifies discrete zeros and continuous intensities, a fixed shared-weight mechanism that emphasizes rare but critical events in both attention and loss computation, and a Gaussian perturbation strategy that enhances learning stability without violating physical constraints. DET further incorporates nonstationary attention to adapt to evolving rainfall regimes. Extensive experiments on multi-decadal South Australian climate data demonstrate that DET consistently outperforms existing deep learning and statistical models across forecasting horizons. Our method provides an effective and generalizable framework for zero-inflated, shift-prone time series, bridging statistical rigor with deep temporal modeling in a unified and scalable design.