I haven't a concrete example but there is a <a href="http://en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29">theorem</a> that says that the derivative of a function has the intermediate value property. This fact makes me think that could be a elementary function without antiderivative.

EDIT:

I am going to be more explicit:

If f is a elementary function, it is defined in the interval (a,b), and it is the derivative of another function (not necessary an elementary function) then f satisfies the intermediate value property inside (a,b).