ListPlot very large numbers

Plot and more generally Graphics can only handle coordinates that are machine numbers, i.e. in the range $MinMachineNumber to $MaxMachineNumber. On my machine these are of the order $10^{308}$.

This limitation is probably present because graphics are rendered by the front end (not the kernel which supports arbitrary precision arithmetic).

The simplest solution would be to linearly rescale everything, i.e. divide the numbers by a large value, before plotting. Then you can manually set the Ticks to reflect the unscaled values.