There are certain integrals that cannot be expressed in terms of elementary functions, which is why special functions have been defined for some of them. The integral of exp(-x2) is (give or take a few factors) defined as the error function. The integral of 1/ln(x) is the logarithmic integral function. Other "impossible" integrals include that of sin(x2) and cos(x2) which are the Fresnel integrals, and integrals that involve the square root of a cubic or quartic polynomial (elliptic integrals).
However, the fact that there exist unmeasurable sets for any measure theory means that there also exist functions that cannot be integrated under any definition of the integral, be that Riemann's definition or even the more advanced Lebesgue integral. Clearly, one cannot hope to integrate the characteristic function of such an unmeasurable set, as the value of its integral would correspond to the measure of that precisely unmeasurable set.