  We construct klt projective varieties with ample canonical class and the
smallest known volume. We also find exceptional klt Fano varieties with the
smallest known anticanonical volume. We conjecture that our examples have the
smallest volume in every dimension, and we give low-dimensional evidence for
that. In order to improve on earlier examples, we are forced to consider
weighted hypersurfaces that are not quasi-smooth. We show that our Fano
varieties are exceptional by computing their global log canonical threshold (or
$\alpha$-invariant) exactly; it is extremely large, roughly $2^{2^n}$ in
dimension $n$. These examples give improved lower bounds in Birkar's theorem on
boundedness of complements for Fano varieties.
