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

Some literature contain some lists of them (e.g., [this paper](https://arxiv.org/abs/1602.06303) ) but most of listed Calabi-Yau threefolds are not simply-connected (or the simply-connectedness is not determined).