  Consider a $p$-random subset $A$ of initially infected vertices in the
discrete cube $[L]^d$, and assume that the neighbourhood of each vertex
consists of the $a_i$ nearest neighbours in the $\pm e_i$-directions for each
$i \in \{1,2,\dots, d\}$, where $a_1\le a_2\le \dots \le a_d$. Suppose we
infect any healthy vertex $v\in [L]^d$ already having $r$ infected neighbours,
and that infected sites remain infected forever. In this paper we determine the
$(d-1)$-times iterated logarithm of the critical length for percolation up to a
constant factor, for all $d$-tuples $(a_1,\dots ,a_d)$ and all $r\in
\{a_2+\dots + a_d+1, \dots, a_1+a_2+\dots + a_d\}$.
  Moreover, we reduce the problem of determining this (coarse) threshold for
all $d\ge 3$ and all $r\in \{a_d+1, \dots, a_1+a_2+\dots + a_d\}$, to that of
determining the threshold for all $d\ge 3$ and all $r\in \{ a_d+1, \dots,
a_{d-1} + a_d\}$.
