Exp(x^2) is a finite composition of a polynomial and an exponential function so from what you posted, it does fall into the Wikipedia definition of elementary function.
They were arguing that the antiderivative F of f: x ↦ exp(x²) is not elementary, whereas f itself is elementary. But both F and f are well-defined functions, which means that f has an indefinite integral, it just happens to be not elementary.
Sometimes Markdown eats following parentheses and sometimes it doesn't, and I've never figured out why. When I first viewed the post it showed me the parenthesis not being eaten, but then it did it anyway when I viewed it from the main page (too late for me to edit). Thanks for fixing it up!