In official tournament chess amateurs can and do play illegal moves constantly. When an illegal move is questioned the rule is that if both players missed the illegal move then the game continues as is.
And in official blitz you can simply capture the king since there's no time to screw around with an opponent who leaves his king hanging.
I've been playing something we call (we invented) "sneaky chess". Basically, it is just like chess except checks aren't announced, there is no checkmate (the king must be taken to win the game), kings can march into check (but don't have to -- stalemate still possible), and the king can remain under check for multiple moves.
Being beginners, we sometimes don't notice that a king is in check for several moves. One game both kings were in check, and the player who first spots it wins (by being the first to take the opponent's king). However, that doesn't happen much any more now that we are more experienced.
What do you think of sneaky chess?