  Call a projective variety $X$ Calabi-Yau if its canonical divisor is ${\bf
Q}$-linearly equivalent to zero. The smallest positive integer $m$ with $mK_X$
linearly equivalent to zero is called the index of $X$. We construct Calabi-Yau
varieties with the largest known index in high dimensions. In our examples, the
index grows doubly exponentially with dimension. We conjecture that our
examples have the largest possible index, with supporting evidence in low
dimensions. The examples are obtained by mirror symmetry from our Calabi-Yau
varieties with an ample Weil divisor of small volume. We also give examples for
several related problems, including Calabi-Yau varieties with large orbifold
Betti numbers or small minimal log discrepancy.
