Here are a few interesting examples. One of the earliest, although not called an adjoint, was the Bohr compactification of an abelian group and the construction was similar to that of the original adjoint functor theorem. The earliest that was labelled such was Kan's Ex^\infty, which was a left adjoint to the inclusion of (what are now called) Kan simplicial sets into simplicial sets. All Galois connections are contravariant adjunctions. The contravariant power set functor is adjoint to itself on the left. Hom(A,PB) = Hom(B,PA). Valid in any topos. Also valid in toposes, for any f: A --> B, the induced inverse image function f*: Sub B --> Sub A has both a left adjoint (the familiar direct image) and a right adjoint (not familiar), considering the subobject lattices as categories. And many, many more.