How can one prove the impossibility of writing $ \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions?

Can we express $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ in terms of elementary functions? (Note: Infinite series are not allowed.) If not, then is there a proof that $ \displaystyle \int e^{x^{2}} \, \mathrm{d}{x} $ cannot be written in such a manner?

This is a simple consequence of the Liouville-Ritt theory of integration in finite terms.

The techniques used for indefinite integration of elementary functions are actually quite simple in this transcendental (vs. algebraic) case, i.e. the case where the integrand lies in a purely transcendental extension of the field of rational functions $\mathbb C(x)$. Informally this means that the integrand lies in some tower of fields $\mathbb C(x) = F_0 \subset \cdots \subset F_n = \mathbb C(x,t_1,\ldots,t_n)$ which is built by adjoining an exponential or logarithm of an element from the prior field, i.e $\ t_{i+1} =\: \exp(f_i)\ $ or $\ t_{i+1} =\: \log(f_i)\ $ for $\ f_i \in F_i$ where $ t_{i+1}$ is transcendental over $ F_i\:.\ $ For example $\ \exp(x),\ \log(x)\ $ are transcendental over $\mathbb C(x)$ but $\ \exp(2\ \log(x)) = x^2\ $ is not. Now, because $\ F_{i} = F_{i-1}(t_{i})$ is transcendental it has particularly simple structure, viz. it is isomorphic to the field of rational functions in one indeterminate $\:t_i\:$ over $\ F_{i-1}\ $. In particular, this means that one may employ well-known rational function integration, techniques such as expansions into partial fractions. This, combined with a simple analysis of the effect of differentiation on the degree of polynomials $\ p(t_i)$, quickly leads to the fundamental result of Liouville on the structure of antiderivatives, namely they must lie in the same field $ F$ as the integrand except possibly for the addition of constant multiples of log's over $ F$. With this structure theorem in hand, the transcendental case reduces to elementary computations in rational function fields. This transcendental case of the algorithm is so simple that it may be easily comprehended by anyone who has mastered a first course in abstract algebra.

On the other hand, the full-blown algebraic case of the algorithm requires nontrivial results from the theory of algebraic functions. Although there are some simple special case algorithms for sqrt's and cube-roots (Davenport, Trager) the general algorithm requires deep results about points of finite order on abelian varieties over finitely generated ground fields. This algebraic case of the integration algorithm was discovered by Robert Risch in 1969 - who did his Berkeley Ph.D. on this topic (under Max Rosenlicht).

For a very nice introduction to the theory see Max Rosenlicht's Monthly paper, available from JSTOR and also here. This exposition includes a complete proof of the Liouville structure theorem along with a derivation of Liouville's classic criterion for $\int f\: e^{g}\: dx\ $ to be elementary, for $\: f,\: g\in \mathbb C(x)$. Namely, the integral is elementary iff $\, f = h'+hg',\,$ for some $\,h\in\Bbb C(x).\,$ Then an elementary intergal is $\,h\,e^g.\,$ For algorithms see Barry Trager's 1984 MIT thesis and Manual Bronstein: Symbolic Integration I: Transcendental Functions.