  Given $r\geq2$ and an $r$-uniform hypergraph $F$, the $F$-bootstrap process
starts with an $r$-uniform hypergraph $H$ and, in each time step, every
hyperedge which "completes" a copy of $F$ is added to $H$. The maximum running
time of this process has been recently studied in the case that $r=2$ and $F$
is a complete graph by Bollob\'as, Przykucki, Riordan and Sahasrabudhe
[Electron. J. Combin. 24(2) (2017), Paper No. 2.16], Matzke
[arXiv:1510.06156v2] and Balogh, Kronenberg, Pokrovskiy and Szab\'o
[arXiv:1907.04559v1]. We consider the case that $r\geq3$ and $F$ is the
complete $r$-uniform hypergraph on $k$ vertices. Our main results are that the
maximum running time is $\Theta\left(n^r\right)$ if $k\geq r+2$ and
$\Omega\left(n^{r-1}\right)$ if $k=r+1$. For the case $k=r+1$, we conjecture
that our lower bound is optimal up to a constant factor when $r=3$, but suspect
that it can be improved by more than a constant factor for large $r$.
