After working 5 years as a developer it is starting to really bug me that my clients (who I work for almost as long) don't take my word for it when I say "that's not possible".

Of course I don't mean that it is impossible per se but rather that this feature costs to much time and therefore money to even research if there is a workaround for it and they would never want it, if they knew the cost. Still, even when I tell them that this way, I have to explain this over 3 meetings with 1-3 days where I am told to try it anyway (which ends with the same arguments I told them in the first meeting) while I could be doing work that really is worth my time and their money.

Shouldn't you be telling them everything is possible, but just charge a lot more money?
You could the meetings and make a lot more money, they asked for it after all, so if it takes longer that is not really your issue ;)

@forkbomber If I would have done that I would have lost them as a client after my first or second commission (I hope that is the right word).

I am not sure how this goes where you work but here in Germany consulting is valued and you are expected to deliver what is best for your customer. So if you just implement something they ask for while knowing that it is unproportional they won't hire you anymore.

Most of your pay here is for your consulting skills. You have to make sure that they are satisfied at the end and not just give them what they think they want.

That's where I come from with my rant. They come back to me everything because they trust me by now and know I advise them properly. So why is it so hard to take my word for it.

(Just to give an example: One time they wanted me to integrate AngularJS into their already finished and running Application, using just jQuery and without changing HTML files)

@Jabalaja makes perfect sense if they are indeed asking the wrong things :)