By a Calabi-Yau threefold, I mean a simply-connected smooth compact K"ahler threefold with trivial canonical class. It has two independent Hodge numbers $h^{1,1}$ and $h^{1,2}$.

What is is the smallest number of $h^{1,1} + h^{1,2}$ for Calabi-Yau threefolds that have been found so far?

Some literature contain some list of them (e.g., this paper ) but most of listed Calabi-Yau threefolds are not simply-connected (or the simply-connectedness is not determined).