TITLE: Natural examples of sequences of adjoint functors
QUESTION [38 upvotes]: 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
$$ ( ...,\Delta,\oplus,\Delta,\oplus,...)$$

REPLY [7 votes]: There are some really nice answers here.  Here's another contribution.
Let [n] denote the totally ordered (n+1)-element set, regarded as a category.  For each positive integer n, we have the usual n+1 order-preserving injections from [n-1] to [n], and the usual n order-preserving surjections from [n] to [n-1].  (I mean the ones used all the time for simplicial anything.)  When you regard them as functors, these injections and surjections interleave to form an adjoint chain of length 2n.