Too long for a comment:
On one hand, I already knew from highschool that $\displaystyle\int e^{x^n}dx$ cannot be written as a combination of elementary functions, unless $n=0$ or $1$. On the other hand, I also knew from the various complex integration techniques that I've studied in college, that just because a function does not admit such a primitive, this does not mean that the value of its definite integral cannot be expressed in a closed form. So, by combining the two, I began to fool around a bit in Mathematica, and plot the graphic of $f(n)=\displaystyle\int_0^\infty e^{-x^n}dx$ for $n>0$ . It decreased abruptly from $f(0)=\infty$ to a minimum of about $0.9$ in $x\simeq2\frac16$ , and then began a slow asymptotic rise towards $f(\infty)=1$ . So I decided to zoom in on $[0,1]$ by changing the variable from n to $\dfrac1n$ , ultimately plotting $F(n)=\displaystyle\int_0^\infty e^{-\sqrt[n]x}dx$. Then I decided to compute some values for a few small natural values of n. I was shocked to see that not only were they exact integers (as opposed to some random transcendental number with a never-ending and non-repeating string of decimals, as I was obviously expecting, given the expression of the function), but they also looked kinda familiar... Hmmm... It was uncanny... :-) This was some time last year, in late $2011$, or early $2012$. Months later, I saw the same figure, $2\frac16$ , mentioned in one of Ramanujan's notebooks. It was a surreal experience... But how Euler arrived at either one of his two famous results (i.e., the infinite product and the integral expression) whole centuries before the dawn of computers, is beyond me... :-)