Game-play Elements that Makes or Breaks a Game

To help with detailing or fleshing out the design of the game, I believe we should create a list of game-play elements that are considered important to us.

So, the question is: what are the major/important game-play elements that makes or breaks a game for you? (Alternatively, what are the major elements that you identify a game with?)

The following is a list of elements I came up with (based on my own personal thoughts/opinion) that are categorized and defined at a level of detail in accordance with my preference. There's possibly categories that I haven't yet considered or thought of yet. If I don't have something on my list that is an important part of a game to you, then you should share it, so that we can consider it for the design of the game. Alternatively, you can share your own model of major game-play elements.

• User Interaction / User Input / Controls - easy to use and feel right.
  
• Ease of Accessibility/Comprehension - that is, easy to get into and play. For example, something that can be understood by simple observation is probably something that is easy to get into. (Game Example: Resonance of Fate's 15 or so tutorials at the beginning resulted in me quitting).
• Atmosphere / Ambience - (Game Example: I thought Shadow of the Colossus created a variety of very nice atmospheres through their design and execution of visuals, camera and sound. Note its use of tension during a boss battle where tension is built up during the puzzle phase where the player has to figure out how to make the boss vulnerable for climbing while being attacked, which is followed by a release of tension during the climbing phase with the aid of music and again there is a release of tension after reaching the top where there is less difficulty for moving about.)
• Comprehensible or Likable Characters/Entities - does the entities' personality, motives, history, etc. resonate with us? Are we able to empathize with the characters? Consider all entities from the protagonist to antagonist to inanimate objects such as the environment (since we tend to personify things).
  
• Story / Exploration - in essence, these are concepts / ideas that are presented or communicated to us through the author / designer.
  
• Logic / Puzzle - is there some (seemingly) comprehensible logic or rule system that incites me to ponder and think?
• Physicality / Power - I'm referring to the pseudo tactile feeling that we get a sense of when using an agile character that moves around or using a powerful character that smashes something solid. (Game Example: just try the characters in Street Fighter 3 or 4.)
• Personalization / Customization / Influence / Control / Building - does our choices matter? Can our decisions lead to unique or different outcomes?
• Experimentation - are there mechanisms in place that allows us to experiment (to satisfy our curiosity)? (Note - this is somewhat related to choices and personalization.)
  
• Intrinsic Rewards / Benefits - does playing or experiencing the game personally improves me as a person in some way? For example, gain a new insight/idea, or improve reflexes, etc.
• Challenge / Achievement / Accomplishment - are their mechanisms for me to test my ability?
  
• Social Elements - anything that promotes social interaction. It can be a controversial topic that provokes social communication, or the support of multiplayer in games, or etc. (Game Example: In Pokemon, part of the motivation for collecting and leveling up is due to the assumption that the pokemons will be used in a battle against another person).