  We show that there are exactly, up to isomorphisms, seven extremal log
Enriques surfaces Z and construct all of them; among them types D_{19} and
A_{19} have been shown of certain uniqueness by M. Reid. We also prove that the
(degree 3 or 2) canonical covering of each of these seven Z has either X_3 or
X_4 as its minimal resolution. Here X_3 (resp. X_4) is the unique K3 surface
with Picard number 20 and discriminant 3 (resp. 4), which are called the most
algebraic K3 surfaces by Vinberg and have infinite automorphism groups (by
Shioda-Inose and Vinberg).
