Self-attention architectures, which are rapidly pushing the frontier in natural language processing, demonstrate a surprising depth-inefficient behavior: previous works indicate that increasing the internal representation (network width) is just as useful as increasing the number of self-attention layers (network depth). We theoretically predict a width-dependent transition between depth-efficiency and depth-inefficiency in self-attention. We conduct systematic empirical ablations on networks of depths 6 to 48 that clearly reveal the theoretically predicted behaviors, and provide explicit quantitative suggestions regarding the optimal depth-to-width allocation for a given self-attention network size. The race towards beyond 1-Trillion parameter language models renders informed guidelines for increasing self-attention depth and width in tandem an essential ingredient. Our guidelines elucidate the depth-to-width trade-off in self-attention networks of sizes up to the scale of GPT3 (which is too deep for its size), and beyond, marking an unprecedented width of 30K as optimal for a 1-Trillion parameter self-attention network.