[CrazyStat]

Precision in p-values that small is more or less meaningless in almost all cases, because any violation of model assumptions will result in p-value imprecision far greater than 10^-10. p-values are (almost always) approximations based on an approximate model, and the variation between the model and reality is probably more than 10^-10.

Some tiny aspect of the real process that your model falls to capture might mean that that 10^-10 is actually 0.001, and 10^-40 is also 0.001. In complex biological fields it's fair to assume that there are always such tiny aspects.

[native_samples]

You're right. The numbers are too small to be plausible. I read on Scott Alexander's blog about 5-HTTLPR that in genetics they can get very low P values relative to most life sciences, but 10^-40 indeed seems far too low for any plausible experiment. I guess even in particle physics they don't go that low.