The World Ends With You for DS does exactly that. You can change the difficulty of the game at any time (except in battles), which affects the items and XP you get from fights. If you lose a fight, you have 3 options: Quit, Retry, and Retry on Easy. Great system.

If you actually read any of the newsposts related to that game, you would have realized that the comic was supposed to be FUNNY, and the Penny Arcade guys (Tycho, at the very least) love the game. It gets my thumbs-up, by the by.

All this discussion and nobody's brought up Phoenix Wright? That series of games is nothing but puzzles, and all of them make sense. I've played through 2 of them, and there was only 1 time in each game that I felt was ridiculously hard (what do you mean I have to look at the painting then the desk then the painting again? There's no clue on the desk that makes me want to look at the painting again! ARGH!) The "puzzles" aren't puzzles in the truest sense of the word, but you do have to think about the topic at hand and decide which piece of evidence would fit best into the situation... isn't that a puzzle?