  Graph bootstrap percolation is a simple cellular automaton introduced by
Bollob\'as in 1968. Given a graph $H$ and a set $G \subseteq E(K_n)$ we
initially "infect" all edges in $G$ and then, in consecutive steps, we infect
every $e \in K_n$ that completes a new infected copy of $H$ in $K_n$. We say
that $G$ percolates if eventually every edge in $K_n$ is infected. The extremal
question about the size of the smallest percolating sets when $H = K_r$ was
answered independently by Alon, Kalai and Frankl. Here we consider a different
question raised more recently by Bollob\'as: what is the maximum time the
process can run before it stabilizes? It is an easy observation that for $r=3$
this maximum is $\lceil \log_2 (n-1) \rceil $. However, a new phenomenon occurs
for $r=4$ when, as we show, the maximum time of the process is $n-3$. For $r
\geq 5$ the behaviour of the dynamics is even more complex, which we
demonstrate by showing that the $K_r$-bootstrap process can run for at least
$n^{2-\varepsilon_r}$ time steps for some $\varepsilon_r$ that tends to $0$ as
$r \to \infty$.
