Number system with $e^x = 0$ for some $x$

If you want the properties $$e^{x+y}=e^xe^y\quad\hbox{for all $x,y$}$$ and $$e^0=1$$ to remain true, then we have $$e^xe^{-x}=1\quad\hbox{for all $x$}$$ and so $e^x$ can never be zero.

Adding to what David said, even if you give up on the homomorphism part of the exponential, the series definition (which holds in any Banach algebra, which includes the number systems) make so that $e^X e^{-X}=1$, so $e^X$ can't be $0$.

The floating-point number system (for a given number of bits) is a finite subset of the extended real number line. It relaxes various algebraic identities so that it can remain closed under as many operations and inputs as possible. As you put it, the "usual properties of arithmetic or the exponential function" are not true in this system, although they are approximately true for the most part.

In this system, $-\infty$ is a number and $e^{-\infty}=0$.

References:
http://pubs.opengroup.org/onlinepubs/9699919799/functions/exp.html
http://en.cppreference.com/w/c/numeric/math/exp