As somebody who's played video games since the day I learned how to hold an x box 360 controller. I can confidently say that I think the deepest connection with friend's is staying up late at night, bullshitting, laughing, and being absolutely exhausted at 2am playing video games.
Whether the night ends with you absolutely crying laughing or your voice just BARLEY above a whisper from being so tired from staying up so late - I don't think there is a more personal and deep moment than staying online with somebody for hours late at night. Even if you LITERALLY just met the person a few days prior and know almost nothing about them