TITLE: Examples for "nice" Boolean algebras that are not complete or not atomic
QUESTION [9 upvotes]: A Boolean algebra may, or may not, be complete (i.e, any set of elements has a sup and an inf) or atomic (i.e., every element is a sup of some set of atoms). 
Boolean Algebras that are complete as well as atomic (also called CABAs) are of course precisely those that are isomorphic to some power set (equipped with the obvious choices for the operations), or equivalently stated, those that form a category dually equivalent to $Set$. 
The category of all Boolean algbras, however, is well-known to be equivalent to the category of Stone spaces (compact totally disconnected Hausdorff spaces) with continuous morphisms. Thus, for a Boolean algebra (of infinite cardinality), it is a very special case to be complete and atomic. My question is: 

What are nice examples for Boolean Algebras that are not complete or not atomic?

Please understand that I do not look for any kind of example (so the emphasis lies on the word "nice"). I am, for instance, aware that looking at free BAs would lead to such an example, and I also know the classic example of the BA that is formed by all finite and confinite sets of integers. Also, as mentioned above, I know how the Stone Duality transforms Stone spaces into Boolean algebras, so please don't simply say "the clopen subsets of a that-and-that Stone-Space form a Boolean algebra".
I admit that nice is a somewhat vague notion. What I mean are Boolean Algebras that arise naturally (except those I have already mentioned) and are of special interest for some reason (yes, I know that this formulation is not vague at all).

REPLY [3 votes]: The 3-volume "Handbook of Boolean Algebras" abounds with examples of Boolean algebras, but of course only few of them are "nice".  

You may have a look at Koppelberg's chapter 6 in volume 1, "Special classes of BAs", and also at several chapters in volume 3 describing some of these classes in more detail.  This may challenge or sharpen your idea of "nice". 
Generalizing your question one could also ask for nice constructions of Boolean algebras (products, free products, quotients, iterated quotients, interval algebras, completions, etc); some of them will yield nice algebras.
For example, dividing any BA $B$ by the ideal $At(B)$ generated by the atoms will yield a new BA $B':=B/At(B)$; this new algebra $B'$ will often be atomless (e.g. if you start with a power set BA, as mentioned by Andreas Blass).  If not, you can continue this process of dividing, even transfinitely often.  
Monk's chapter 12 in volume 2 describes several methods for getting "many" BAs with particular properties.  For example, starting with any linear order $L$, the interval algebra $I(L)$ is defined as the subalgebra of $P(L)$ (the power set algebra) generated by the half open intervals $[a,b)$ in $L$. Sufficiently different/nice linear orders will give nonisomorphic/nice BAs.
For $L=\mathbb N$ you will get the (highly atomic) finite-cofinite subalgebra of $P(\mathbb N)$, and for $L=\mathbb Q$ you will get the countable atomless BA, mentioned by Joel Hamkins.