I am looking for examples of sequences of adjoint functors. That are (possibly bounded) sequences $$(...,F_{-1}, F_{0}, F_1, F_2,...)$$ such that each $F_n$ is left adjoint to $F_{n+1}$. We call such a sequence cyclic of order $k$ if for one $n$ (and hence for all) we have $F_{n} \cong F_{n+k}$. It is relatively easy to prove that cyclic sequences of all orders and non-cyclic sequences of all possible length exist. This can e.g. be done using posets, see http://www.springerlink.com/content/pmj5074147116273/.

I am looing for more "natural" examples of such sequences that are as long as possible. By natural I mean that they grow out of "usual functors" (sorry for this vague statement...)

Let my give two short examples: 1) Let $U: Top \to Set$ be the forgetful functor from locally connected topological spaces to sets. This induces a sequence of length 4: $$ (\pi_0 , Dis , U , CoDis) $$ where $Dis$ and $CoDis$ are the functor that equip a set with the discrete and indiscrete topology. Then the sequence stops. Tons of examples of this type are induced by pullback functors in algebraic geometry.

2) a cyclic sequence of order 2: the Diagaonal functor $\Delta: A \to A \times A$ for any abelian category $A$ is left and right adjoint to the direct sum $$ ( ...,\Delta,\oplus,\Delta,\oplus,...)$$