I'm looking into Lawvere's formulation of unities of opposites and opposites of unities, and for this I would be interested in systems of five (or more) adjoint functors $X\stackrel{\stackrel{\xrightarrow{a}}{\xleftarrow{b}}}{\stackrel{\xrightarrow{c}}{\stackrel{\xleftarrow{d}}{\xrightarrow{e}}}}Y$, ideally within the domains of topos theory and algebraic geometry, but really any examples would be interesting. I know they are very rare, but I feel like I have come across one at least once. Anyone?