Most of the examples of adjoint functors I saw ''in the wild'' have a right adjoint forgetting a part of structure, and left adjoint recovering it in the most efficient/general way. Often, a functor forgetting the part of the structure will be immediately obvious, but we will go through some work to construct the left adjoint for it (i.e. free groups, induced representations).
I'm curious: is there a case when we are interested in finding a right adjoint to some known functor?
Yes, there are many examples. For example, consider the forgetful functor $\mathsf{Grp} \to \mathsf{Mon}$. It is cocontinuous, hence has a right adjoint (SAFT). You can write it down, it sends a monoid to its group of units.
In general, a subcategory of a category is called coreflective if the inclusion functor has a right adjoint (called coreflector). For example, the category of torsion abelian groups is a coreflective subcategory of the category of abelian groups. The coreflector sends an abelian group to its torsion subgroup.
If $R \to S$ is a ring homomorphism, then the forgetful functor $\mathsf{Mod}(S) \to \mathsf{Mod}(R)$ is cocontinuous, hence has a right adjoint. It is given by $\underline{\hom}_R(S,-)$. A similar construction works for finite morphisms $f$ of schemes, where one finds a right adjoint to the functor $f_*$ between quasi-coherent sheaves (which has left adjoint $f^*$). The corresponding question in the derived category yields to Serre duality as explained here.