Calculators can only work to a certain number of decimal places. When you ask the calculator to evaluate $\displaystyle (e^x-1-x)^x$ with $\displaystyle x=10^{-9}$ say, it starts by evaluating $\displaystyle e^x-1-x$ for that value of x. But $\displaystyle e^x = 1+x+\tfrac12x^2+$ higher powers of x, so $\displaystyle e^x-1-x $ is approximately $\displaystyle x^2/2$, which in this case will be less than $\displaystyle 10^{-18}$. That is probably too small a number for the calculator to store, so the poor machine suffers from "

arithmetic underflow" and replaces this quantity by 0. Then when 0 is raised to any power, it is still zero, of course. So the calculator gives the result of the calculation as 0.

If you start with somewhat larger values of x, say $\displaystyle x=10^{-1},\ 10^{-2},\ 10^{-3},\ \ldots,$ you should find that the values that the calculator gives for $\displaystyle (e^x-1-x)^x$ get closer and closer to 1, until at a certain stage arithmetic underflow sets in and the values suddenly become 0.