  By Hacon-McKernan, Takayama, and Tsuji, there is a constant r_n such that for
every r at least r_n, the r-canonical map of every n-dimensional variety of
general type is birational. We give examples to show that r_n must grow faster
than any polynomial in n.
  Related to this, we exhibit varieties of several types (Fano, Calabi-Yau, or
general type) with small volume in high dimensions. In particular, we construct
a klt n-fold with ample canonical class whose volume is less than 1/2^(2^n).
The klt examples should be close to optimal.
