They do exist
I used to think I was an integration hot-shot until I came across e^(e^x). Go on, have a go. You can throw everything you've got at it - substitution, integration by parts... - but you just make it worse, like picking at a scab.
Please note my correct use of the ellipsis.
You CAN, however, approximate a solution to this bastard by using a Taylor expansion.
Another impossible integral: 1/(ln(x))
A possible but pretty tough one, good practice: sqrt(tan(x)). And remember, you don't learn anything by asking Maple.
later: Correction, you learn that Maple makes a big ugly mess attempting to integrate sqrt(tan(x)). Ask it to simplify(%); its answer for a laugh.