Given the formula for the standard normal distribution:
$$ \frac{1}{\sqrt{2\pi}} e^{-\frac{1}{2}x^2} $$
There's clearly an exponential in there. Both the normal and the exponential distributions seem to "trail off" towards $\pm \infty$, and I have an intuition that the normal distribution approaches an exponential distribution towards $\pm \infty$ but I'm not sure how to either formalize or disprove that intuition, whichever is appropriate.
My thinking is that some populations which seem normally distributed (chess ELO ratings could be a rough example, the approximate distribution of which can be seen here) might start to appear exponentially distributed towards the best of the best (or the worst of the worst). Intuitively this seems correct, since the best player in the world might be dozens of points above the second best in the world, but the median player would be tiny fractions of an ELO point above the next best player. It often appears as though the best player is some multiple better than the second best player, and the second best player is also some multiple better than the third best player.
I realise these ideas are not very formal and I'm relying a lot on personal experience. I want to either 1) figure out some formal reason behind why the normal would approach an exponential distribution towards $\pm \infty$, or 2) come up with a proof for why this is not the case.