1. Yes, I want enemies that are theoretically impossible to beat.
2. Yes, I want enemies that are indeed impossible to beat.
3. No.
Because a lot of people in this thread seem to have misunderstood "impossible" as "almost impossible" or even just "very hard".
Of course this is something that I would want too, as I like having the odds against me and still prevailing (the tougher the fight, the better the feeling of winning).
But it is not what I really expected from the description of this thread.
Take gods for example. Remember Vivec from Morrowind? I'm assuming he is the kind of "impossible" that people understand here.
One would initially assume it's impossible to kill a god like him, yet it wasn't much of a hassle in the end, with the right approach and maybe some luck.
My point is: there should be someone or something that can always put the player character in his place, if the he provokes it so. Even if it's the god themselves.
Naturally, the player should have to realize that it's a mistake taking them on, and should have both an option to challenge them and a very easy second option to not do so.