What game most affects the way you play?
I think there are several games that have impacted my roleplaying. The first would be the BECMI Basic Dungeons and Dragons or to be clearer. B C I as my parents did not get the set in order. As such I had to repeatedly fill in the Expert set content of my own devising. We never really got past Expert in actual play, but the concepts in Companions and Immortals were tantalizing. The fact that I had huge gaps in the available content (Being in Korea on the army base at the time) I was forced to create my own content. I remember at one time having a superman class on a college rule sheet of paper, even if not used it was formative to my desire to create original content.
The next roleplaying game that affected me was West End Games Star Wars. During the satanic panic for D&D and us moving back to the United States all the D&D stuff was throw away. My brother and I moved on to Star Wars and enjoyed that for many years. The more freeform type of play that was focused on story was a change. The main rule book introduced me to the concept of “In Media Res”. The concept that many things happened previously “off screen” as opposed to always starting at the beginning helped tremendously in crafting stories.
Other formative sources were in the types of stories I love even if not RPG’s. Movies like Nausicaa (the bad dub at first), Lensmen (a bad adaptation), Star Wars, Labyrinth, Dark Crystal, and others; shows like MacGyver, A Team, Star Trek and other; books like the Earthsea novels, Lord of the Rings, and the Dragonriders of Pern among others.