  The experiments at the LHC are searching for many different final states that
can hint to the presence of new physics beyond the Standard Model. One of the
most interesting and promising sectors for these searches is that of the top
quark, for both theoretical and phenomenological reasons linked to its large
mass and to its possible special role in the electroweak symmetry breaking
sector. We suggest that multi-top events, beyond the standard $t$-$\bar t$ and
four top searches, can bring further insight in constraining and discovering
physics beyond the Standard Model, taking advantage of experimental techniques
similar to those used in present top-quark analyses. This is relevant both for
the next data taking runs at the LHC and even more at higher luminosity and
higher energy collider options, which are discussed for future LHC upgrades and
future accelerators. In particular we consider six top and eight top final
states, discussing the generic colour representations for beyond the Standard
Model particles giving rise to those final state. We also discuss the limits
which can be extracted by using the present analyses sensitive to four top
final states, as well as the potential bounds from new searches we propose to
experimental collaborations as an alternative.
