@OP: I'll go with the "easy to learn, hard to master" approach. There are lots of games where this applies, but also games where it simply doesn't...
-games where you level up or gain newer abilities during the game shouldn't be counted. Sure, it looks as if you improved, but this method is actually a bit like cheating. In a good game, all the rules should be layed out at the beginning. This isn't to say you can't get better (checkers both utilises simple and complex, despite your pieces "levelling up" if they reach the other side of the board), but it should be in consideration. There shouldn't be any grind to it.
-there are also games that already start out pretty complex or hard. Most RTS-es have this whole tech tree you'll have to know or find out, shooters have these maps and styles. Open world games often are a toolbox of all sorts of games (though they're called "minigames" nowadays). I could go on.
Man...now I think of it, it'll be hard to name anything other than platformers, board games or puzzle games to fill the "easy to learn, hard to master" criteria.
Of course, not all games should try to incorporate both. It's perfectly fine to have a game that tutorializes people in the course of several hours. And games that are simple but have no depth to them...hmm...okay, I'll admit to that: if you want to make a simple game, make sure there is a depth to it; something to master or perfect. If not (or if the skill isn't gradually learned through the game) it'll be just a boring game.
There is certainly something to be said about the simplicity & complexity of single game characters, but I'd rather keep that for a different topic. I don't want to sound like the guy who wants to deem his "gameplay is all that matters!!!!" standpoint, but gameplay and story can't be measured by the same standards. and comparing characters, that should be done with other characters*. I mean...you won't think less of chess because the pieces don't have names or deep motivational motives for the moves you're making them do, right?
Now for a recent example...Zuma's revenge. Things really can't start out more simple: you're at the end of a sort of railroad track that circles around you and colored balls head toward you. Your only weapon is shooting colored balls at that track; match three or more of the same color and that part will disappear. In the end, you have to avoid the line of balls to make it all the way to your position. It's really as simple as it sounds (and looks even easier if you see it in action), but that doesn't make it a simple game. Oh, and the plot is virtually non-existent as well. You wash up on some island and some bad guys don't like it. You fight them. It's not going to win prizes in that aspect but I'd rather keep it like this than pitiful attempts to shoehorn in unnecessary drama (you don't make that shit up when playing sports either, right?).
That said...the game is far more complex than it looks at first glance. For one thing, the string of balls that come your way isn't endless. It depends on the score. Which means that striking combo's (which award more points), shooting through self-made openings for combo's behind the front lines and catching the bonusses that appear on the side of the tracks actually matter for other things than cosmetic reasons (the more you have of those, the faster the constant supply of balls stops). Chaining combo's tend to create more one-time bonusses on the track and those are things you need to rely on in the higher levels. So while quick calculations of where to shoot your balls still matter, thinking ahead (while under constantly building pressure) is important as well.
*as it stands, I'm compiling my personal list with favorite villains. It contains both movies, series, games and comic books. there, the PC characters make their appearance DESPITE being on the PC
