Could somebody recommend a book or an online resource where I could read up on the proof that normal CDF has no close-form expression?
1. It does in the form of the error function, it does not have a closed form representation as a finite combination of algebraic operations and elementary functions.
2. You need to look up Liouville's Principle. you could star here and follow the links
3. Most (indefinite)integrals of elementary function have no closed form representation as a finite combination of algebraic operations and elementary functions.
RonL
It results from (or at least is closely related to) Liouville's theory. You should have a look there. I don't think it answers your precise question, but it gives precise definitions and other examples (with proof) of non elementarily integrable functions, like $\displaystyle \frac{e^x}{x}$, which should give you an idea of the kind of arguments involved.
And there are references inside as well. For instance, if you study/work in a university, you may probably easily find the following article in your maths library:
American Mathematical Monthly Februrary 1961 "Integration" by D.G. Mead p 152-156.
You may be interested to look at differential Galois theory in you want to learn more.