Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that happen to have trivial class group, localizations of these, and completions of localizations of these at a prime. Are there more exotic examples? Is there anything like a classification? 



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](http://en.wikipedia.org/wiki/B%C3%A9zout_domain), 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](http://alpha.math.uga.edu/%7Epete/integral.pdf). (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](https://projecteuclid.org/euclid.dmj/euclid.dmj/1077310578): 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.


