I'm almost certain that the answer is no (I'm sure others will post with reasons why in due course). Your best bet is probably going to be on a flight outside the EU. Last year I flew wizair from Poland to Kiev. Once we were out of EU the pilot seemed happy to let a couple of enthusiasts up onto the flight deck and they remained there until after landing. Maybe you could get I touch with them and try your luck.
It's a shame that such a request is such a big deal these days especially, as you say, it costs so much to train. A lot of money to throw away if you find you don't like it on your first day at the office!