  Given a fixed graph $H$ and an $n$-vertex graph $G$, the $H$-bootstrap
percolation process on $G$ is defined to be the sequence of graphs $G_i$,
$i\geq 0$ which starts with $G_0:=G$ and in which $G_{i+1}$ is obtained from
$G_i$ by adding every edge that completes a copy of $H$. We are interested in
the maximum number of steps, over all $n$-vertex graphs $G$, that this process
takes to stabilise. In the first of a series of papers exploring the behaviour
of this function, denoted $M_H(n)$, and its dependence on certain properties of
$H$, we investigate the case when $H$ is a cycle. We determine the running time
precisely, giving the first infinite family of graphs $H$ for which an exact
solution is known. The maximum running time of the $C_k$-bootstrap process is
of the order $\log_{k-1}(n)$ for all $3\leq k\in \mathbb{N}$. Interestingly
though, the function exhibits different behaviour depending on the parity of
$k$ and the exact location of the values of $n$ for which $M_H(n)$ increases is
determined by the Frobenius number of a certain numerical semigroup depending
on $k$.
