The Moore bound sets limits on this, but for many years mathematicians in the field have been interested in a more precise answer.
The table below gives current progress on this problem (excluding the case of degree 2, where the largest graphs are cycles with an odd number of vertices).
Only a few of the graphs in this table (marked in bold) are known to be optimal (that is, largest possible).
The remainder are merely the largest so far discovered, and thus finding a larger graph that is closer in order (in terms of the size of the vertex set) to the Moore bound is considered an open problem.
In all other cases, the footnotes in the table indicate the origin of the graph that achieves the given number of vertices: