  A non-nilpotent variety of algebras is almost nilpotent if any proper
subvariety is nilpotent. Let the base field be of characteristic zero. It has
been shown that for associative or Lie algebras only one such variety exists.
Here we present infinite families of such varieties. More precisely we shall
prove the existence of
  1) a countable family of almost nilpotent varieties of at most linear growth
and
  2) an uncountable family of almost nilpotent varieties of at most quadratic
growth.
