I found somewhere on the internet that $e$ can be approximated by $$\Large\left(1+9^{-4^{7\times6}}\right)^{3^{2^{85}}} $$
Note that the expression uses each numerical digit exactly once.
I understand that this works because $$e=\lim_n \left(1+\frac{1}{n}\right)^n.$$
Using the same idea I got another approximation to $e$
$$\large\left(1+\left(2^{3}\right)^{(4+5)(6-7)}\right)^{8^9} $$
I know this approximation is not nearly as good as the previous one, but it uses the digits in order. (Is it the best possible?)
Do these types of expressions have a name?
They are called pandigital expressions (Thanks Wojowu)
Are there similar expressions for $ \pi$ or the golden ratio or any other interesting number?
Is there a way of constructing this type of expressions for any given integer, rational or real ?
edit: (Inspired by Arthur's comment) Are there expressions similar to these not using base ten?
You can extend your idea to higher bases to get as accurate as you like. For example in base $N=2^{3456789}+1$:
$$\Large{\left({1+(2^{3456789}})^{(10-11)^{(-12+13)\cdots(-(N-2)+(N-1))}}\right)}^{N}$$