Varieties of general type with doubly exponential asymptotics

By a theorem of Hacon–McKernan, Takayama, and Tsuji, for every n there is a constant r_n for which every smooth variety X of dimension n of general type has birational pluricanonical maps |mK_X| for m ≥ r_n.  In joint work with Burt Totaro and Chengxi Wang (see  https://arxiv.org/abs/2109.13383), we show that the constants r_n grow at least doubly exponentially.  Conjecturally, it's expected that the optimal bound is in fact doubly exponential.  We do this by finding weighted projective hypersurfaces of general type with extreme behavior: this includes examples of very small volume and many vanishing plurigenera.  We also consider the analogous questions for other classes of varieties and provide some conjecturally optimal examples.  For instance, we conjecture the terminal Fano variety of minimal volume and the canonical Calabi-Yau variety of minimal volume in each dimension.