Are there known pairs of numbers $a$ and $b$, which at first look at them seemed likely to be equal, and after checking up to $10^n$ decimal places appeared to agree, but suddenly for some $n$ they would appear unequal?
Of course, I don't mean such things like $\arccos(-1+10^{-10000})\approx\pi$, which are clearly ad hoc and uninteresting. I mean something more like $\pi^4+\pi^5\approx e^6$ (which holds with relative error $\approx 4\times10^{-8}$), but would like to have some similarly simple approximate equality valid with relative error of $10^{-100}$ or even smaller.
$e^{\pi\sqrt{163}}\approx 262\,537\,412\,640\,768\,744$ to 12 decimal places.