Playing games won't make you a good designer, understanding the constraints within which actual development occurs will. Thinking through your ideas in terms of actual gameplay will make you a better designer, thinking up "cool ideas" will not.
Playing games should give you interesting points of reference when designing a similar game, but theory and goals of design are more important than playing games, since you probably don't know "why" a design is good, so playing Final Fantasy VII is not going to teach you anything in that regard. If I was going to teach people design, I would actually have them read books on design, get them to build games with post it notes and collaborate with people outside their field of expertise. Also you have to be able to present your idea to other people and allow them to tear it apart.
Penny Arcade has done some relatively good points for people who have some interest in this;
http://www.penny-arcade.com/patv/episode/so-you-want-to-be-a-game-designer
http://www.penny-arcade.com/patv/episode/playing-like-a-designer-pt.-1
Since you posed this as a question rather than simply a request for a list, I am providing my opinion based on some experience. I think you're not required any one game, just have a good breadth of experience in gaming.
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