  We construct smooth projective varieties of general type with the smallest
known volume and others with the most known vanishing plurigenera in high
dimensions. The optimal volume bound is expected to decay doubly exponentially
with dimension, and our examples achieve this decay rate. We also consider the
analogous questions for other types of varieties. For example, in every
dimension we conjecture the terminal Fano variety of minimal volume, and the
canonical Calabi-Yau variety of minimal volume. In each case, our examples
exhibit doubly exponential behavior.
