How can I find the value of a non-repeating exponent tower?

Question

There are ways to express the function $$f(x)=x^{x^{x^{x^{x^{x^\cdots}}}}}$$ with $f(x)=\dfrac{W(-\ln(x))}{-\ln(x)} $ for other function like this; $$g(x) = x^{-x^{x^{-x^{x^{-x^{x^\cdots}}}}}} $$ I don't know how to express in a simple function, but I can put it like. $$ \exp\left({\frac{W(-\ln(y^y))}{y}}\right) = x$$ But for something that doesn't repeat like. $$h(x)= x^{-x^{-x^{x^{x^{x^{-x^{-x^{-x^{-x^\cdots}}}}}}}}} $$ How could I write it as an implicit function? Or find the value of $h(e)$?