Exotic principal ideal domains

No, to the best of my knowledge there is nothing like a general classification of PIDs. Despite their easy definition, they turn out to be rather a finicky class of rings, as for instance Gauss conjectured that there are infinitely many PIDs among rings of integers of real quadratic fields, but more than $200$ years later we have not been able to prove that there are infinitely many PIDs among rings of integers of all number fields. And, as came out in the comments to Emil's answer, the property of being a PID is not first order, so is not very robust in a model-theoretic sense. In that regard, the better class of rings are the Bézout domains, i.e., domains in which every finitely generated ideal is principal. A theorem of Kaplansky which can be used to show that various "big" domains (e.g. $\overline{\mathbb{Z}}$, the ring of all algebraic integers) are Bézout can be found at the end of the section on overrings in these notes. (I am now giving less precise citations to my often-changing commutative algebra notes in the hope that they will take longer to become obsolete.)

There are some interesting papers on construction of PIDs with various properties. The one I want to read next is this 1974 paper of Raymond C. Heitmann: given any countable collection $\mathcal{F}$ of countable fields containing only finitely many fields of any given positive characteristic, Heitmann constructs a countable PID of characteristic $0$ with residue fields precisely the elements of $\mathcal{F}$.

Added: note that $\overline{\mathbb{Z}}$ is also an antimatter domain, i.e., it has no irreducible elements (which specialists in the field tend to call "atoms"). Thus this gives an example of a Bézout domain which is not an ultraproduct of PIDs.

Smith constructed a PID which is a nonstandard model of open induction. That should be exotic enough. (Note that nonstandard models of just slightly stronger theories of arithmetic, such as $IE_1$, are never even UFDs.)

Fontaine's ring $B_{cris}^{\varphi=1}$ is a PID, and no expert in the field would have bet on it in the first place (this led to some very nice recent developments by Fargues and Fontaine).

http://www.math.u-psud.fr/~fargues/Courbe.pdf