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I fall somewhere between Story and Hardcore. If I like a game, I will usually do a good chunk of the side content. I have never platinumed a big game though. I complete the story of most games, unless they don't grab me and I quit within the first several hours. The amount of side content I do directly relates to 1) how much I like playing the game, and 2) if I have something else that I am really looking forward to playing. I don't care about trophies or gamerscore at all.
Edit to add: I generally try to do a lot of side content while working on the story. In most games, once I'm done with the story, I move on to something else. So, in order to stretch the games out, I do both at the same time.
One recent exception to this was God of War. I did tons of side content after finishing the story. But, that was an exceptionally good game.
