I am interested in what is known about complete Boolean algebras $B$ with the countable chain condition (ccc), i.e., every disjoint set is countable. Let $K$ be the Stone space of $B$; the completeness of $B$ is equivalent to the fact that $K$ is Stonean (extremally disconnected compact Hausdorff).

Let $K_d$ be the closure of the isolated points of $K$ and denote the complement by $K_c$, so $K = K_d \sqcup K_c$; ccc implies that there are countably many isolated points. On the other hand, [Banach spaces of continuous functions, Theorem 5.1.3] (originally by Dixmier) shows that $K = K_h \sqcup K_a$ where $K_h$ is hyperstonean and $K_a$ has no nontrivial normal measures, and $K_a = K_m \sqcup K_n$ with $K_m$ containing a dense meagre subset and in $K_n$ every meagre subset is nowhere dense and the support of every measure is nowhere dense. Denote the measure algebra of $[0,1]$ (Lebesgue measurable sets modulo null sets) by $\Sigma$. By Maharan's Theorem $K_h = K_d \sqcup K_\Sigma$ where $K_\Sigma$ is either the Stone space of $\Sigma$, or empty. (All these notations were made up by me.) It follows that
$$
K = K_d \sqcup K_\Sigma \sqcup K_m \sqcup K_n
$$
The unique atomless separable complete Boolean algebra (see [Banach spaces of continuous functions, Theorem 1.7.11] or https://mathoverflow.net/questions/470465/description-of-atomless-complete-boolean-algebras-with-a-countable-pi-base) has a Stone space homemorphic to the Gleason cover of $[0,1]$ which has no normal measures, so if it exists, it is contained in $K_a = K_m \sqcup K_n$. In which part is it contained? ([Banach spaces of continuous functions, Example 5.1.4] does mention this fact about the Gleason cover but fails to answer this question.) What else is know about $K_a = K_m \sqcup K_n$ or equivalently, its Boolean algebra counterparts? Is some kind of characterization known?