  We completely solve the problem of enumerating combinatorially inequivalent
$d$-dimensional polytopes with $d+3$ vertices. A first solution of this
problem, by Lloyd, was published in 1970. But the obtained counting formula was
not correct, as pointed out in the new edition of Gr\"unbaum's book. We both
correct the mistake of Lloyd and propose a more detailed and self-contained
solution, relying on similar preliminaries but using then a different
enumeration method involving automata. In addition, we introduce and solve the
problem of counting oriented and achiral (i.e. stable under reflection)
$d$-polytopes with $d+3$ vertices. The complexity of computing tables of
coefficients of a given size is then analyzed. Finally, we derive precise
asymptotic formulas for the numbers of $d$-polytopes, oriented $d$-polytopes
and achiral $d$-polytopes with $d+3$ vertices. This refines a first asymptotic
estimate given by Perles.
