You can build long chains of adjoints by taking functor categories and using Kan extensions.  I will give an example.

Write $\underline{n}$ for the set $\{1, \ldots, n\}$ considered as a discrete category.  As $\underline{1}$ is the terminal category, there is a unique functor $f \colon \underline{2} \to \underline{1}$.

Precomposing with $f$ gives a functor $f^* \colon [\underline{1}, \mathrm{Set}] \to [\underline{2}, \mathrm{Set}]$.  As a functor $\underline{n} \to \mathrm{Set}$ is just an $n$-tuple of sets, we have that $f^*(X) = (X, X)$ so $f^*$ is the diagonal functor.  The functor $f^*$ has adjoints on both sides, which are the Kan extensions.  The left Kan $f_!$ provides a method of combining two sets $f_!(X,Y)$ so that a map $f_!(X,Y) \to Z$ is the same data as a pair of maps $(X \to Z, Y \to Z)$.  In other words, $f_!(X,Y) = X \sqcup Y$ is the disjoint union.  Similarly, the right Kan $f_*(Y,Z)$ provides a method of combining $Y$ and $Z$ so that $X \to f_*(Y,Z)$ is the same data as a pair of maps $(X \to Y, X \to Z)$.  In other words, $f_*(Y,Z) = Y \times Z$.  So far, we have a two-step adjunction of known useful functors
$$
f_! \dashv f^* \dashv f_*
$$

Precomposing with $f^*$ gives a functor $(f^*)^* \colon [[\underline{2},\mathrm{Set}], \mathrm{Set}] \to [[\underline{1},\mathrm{Set}], \mathrm{Set}]$.  As a functor $[\underline{2},\mathrm{Set}]\to \mathrm{Set}$ is an operation $\odot \colon \mathrm{Set} \times \mathrm{Set} \to \mathrm{Set}$ that takes two sets and returns a set, we have that $(f^*)^*(\odot)$ is the "squaring" functor $(X) \mapsto X \odot X$.

Certainly we have Kan extensions
$$
(f^*)_! \dashv (f^*)^* \dashv (f^*)_*
$$
However, if $L \dashv R$ is an adjunction, we always have $L_! \cong R^*$ and $L^* \cong R_*$.  Consequently, we can build the five-functor chain
$$
(f_*)_! \dashv (f_*)^* = (f^*)_! \dashv (f^*)^* \dashv (f^*)_* = (f_!)^* \dashv (f_!)_*
$$

For example, the functor $(f^*)_!$ takes a unary operation $\Gamma \colon \mathrm{Set} \to \mathrm{Set}$, considers it as a functor defined on the diagonal $\mathrm{Set} \overset{\Delta}{\to} \mathrm{Set} \times \mathrm{Set}$, and freely generates from here a binary operation.  By the formula $L_! \dashv R^*$, we can calculate this rather concretely as sending $\Gamma$ to the binary operation $X \otimes_\Gamma Y = \Gamma(X \times Y)$.  A natural transformation of bifunctors $\otimes_\Gamma \implies \odot$ will somehow be the same information as a natural transformation $\Gamma \implies (X \mapsto X \odot X)$.

We could evidently continue this pattern to build even longer and less-understandable chains.