Let $X$ be a compact Hausdorff topological space whose convergent sequences are eventually constant. Is there a description of such spaces. How ''far'' these spaces from Stonean ones?
Norbert, I don't think that there is a reasonable description of such spaces. Indeed, they properly contain spaces $K$ for which the Banach space $C(K)$ is Grothendieck and this class is far from being fully delineated. For example, there are connected, compact spaces $K$ for which $C(K)$ is indecomposable, hence Grothendieck.