Maybe, they're meant to be fun?
I dunno, i think that the essence of a game is to be fun. And fun can be found in sadness, drama, horror, circus (clown) and even work. Fun is about enjoying.
And, I want to know more about the multiple levels you said.
http://fr.wiktionary.org/wiki/fun (in french)
Fun for me is "divertissement" (entertainment), games are entertainment.
Game is about playing, it's about entertainment. You know, art isn't a form of elitism, it's just cultural entertainment.
Games are entertainment and a form of art. Like cinema is entertainment, painting is entertainment, and all should be about entertainment.
Oh guys, please, don't want to explain the difference between Chicago's and Frankfurt's schools point-of-view, but clearly, i'm one of the guys that see entertainment as art, and art as entertainment. And for me, games are about playing, playing is about entertainment and should be FUN.
Ok?
And by the way:
Etymology of the word "game" by wiktionary
From Middle English game, gamen, gammen, from Old English gamen (sport, joy, mirth, pastime, game, amusement, pleasure), from Proto-Germanic *gamanan (amusement, pleasure, game", literally "participation, communion, people together), from *ga- (collective prefix) + *mann- (man), equivalent to ge- +‎ man; or alternatively from *ga- + a root from Proto-Indo-European *men- (to think, have in mind), equivalent to ge- +‎ mind. Cognate with Middle High German gamen (joy, amusement, fun, pleasure), Swedish gamman (mirth, rejoicing, merriment), Icelandic gaman (fun). Related to gammon, gamble.