Your boss comes to you and says, "How would you like to play games all week instead of your normal work?" After several seconds of blank stare followed by a thought that this might be his good twin, you readily accept the offer, running home to bring back a trunk-full of your favorites.
Instead, let's say he made you this offer. "I have 500 endgame positions from Torres. It's your final move. Over the course of the week, you must determine if you can win each game with your last actions." I don't know about you, but I would not take this offer.
A large part of the enjoyment I get from gaming is playing by intuition. I have a sense of what's going on, what I should be doing, and my scoring position. I always resist the urge to "count things out".
Some games are very conducive to calculation: Alhambra, Carcassonne, El Grande, Medina, Through the Desert, Tikal, and Torres. But even in these, I avoid constant analysis. I don't care much if others do it, as long as they don't take an unreasonably long time to move.
Even in 100% information pure abstract games (e.g. Chess, Go, Gipf series), I do not memorize openings or look ahead 20 moves. It's much more fun to have the "let's see what this does" style of play.
Of course, as you gain experience in a game, you will start to make better moves based on past failures. This internalized calculation may look like analysis, but it is really intuition. That is the key. Intuition grows as you learn the game.
For example, in Go you first learn about life and death. You may not see it happening until the final stone is played in that area of the board. Then as your intuition grows, you'll see a position and know it is alive without playing it out (even in your mind). Now you can start to apply your intuition to deeper plays potentially connected to this "live" group.
When seemingly deep and calculating game play becomes second nature, the play of a game takes on a new level. This is always something worth attaining.
However, intuition is not simply repetition of experience. Our minds have the phenomenal ability to generalize patterns and to make cognitive leaps without any conscious rationale or inference. For me, this is fun and satisfying.
[Torres image by Kimball Bent.]