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Consider a $p$-random subset $A$ of initially infected vertices in thediscrete cube $[L]^d$, and assume that the neighbourhood of each vertexconsists 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 weinfect 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 aconstant 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 forall $d\ge 3$ and all $r\in \{a_d+1, \dots, a_1+a_2+\dots + a_d\}$, to that ofdetermining the threshold for all $d\ge 3$ and all $r\in \{ a_d+1, \dots,a_{d-1} + a_d\}$.

The $d$-dimensional bootstrap percolation models with threshold at least double exponential