  In this paper we construct infinitely many examples of toric Fano varieties
with Picard number three, which do not admit full exceptional collections of
line bundles. In particular, this disproves King's conjecture for toric Fano
varieties.
  More generally, we prove that for any constant $c>\frac34$ there exist
infinitely many toric Fano varieties $Y$ with Picard number three, such that
the maximal length of exceptional collection of line bundles on $Y$ is strictly
less than $c\rk K_0(Y).$ To obtain varieties without exceptional collections of
line bundles, it suffices to put $c=1.$
  On the other hand, we prove that for any toric nef-Fano DM stack $Y$ with
Picard number three, there exists a strong exceptional collection of line
bundles on $Y$ of length at least $\frac34 \rk K_0(Y).$ The constant $\frac34$
is thus maximal with this property.
