What are largest betti numbers $b_2$ and $b_3$ of three-dimensional Calabi-Yau manifolds that are discovered for today?
Is there some nice reference?
Since mirror symmetry exchanges the Hodge numbers $h^{1,1} = b_2$ and $h^{2,1} = \frac{1}{2}(b_3 - 1)$, it is perhaps more natural (and of course equivalent) to discuss these. The record-holders all come from the list of hypersurfaces in toric fourfolds, constructed by Kreuzer and Skarke. The largest value of $h^{1,1}$ is $491$, and the same for $h^{2,1}$. These two manifolds (which are mirror) also hold the record for largest $h^{1,1} + h^{2,1}$, which is 502. There is a third manifold which shares this record; its Hodge numbers are $(251,251)$, and it is also in the Kreuzer-Skarke list.
See this recent paper and this one for nice plots and a discussion of these Calabi-Yau threefolds and others with large Hodge numbers.