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.