Re: Washing a baseball cap........
 

Imo that is not a stain. That is colour fading/changing/discolouration in the fabric itself, maybe due to sweat, sun, and possibly oils from the fingers and rubbing (which causes breakdown in fibres) from said friend always removing the hat with the left hand in the exact same spot.

Is this a heavily worn hat?

If it's fading/breakdown, you will have to learn to love it this way or attempt a hazardous redye (after carefully covering the logo with tape) with something like this.

https://www.amazon.com/Tulip-ColorSh...024479508&th=1

You could use a grey fabric marker if you just want it to be grey, but the odds of getting a colour match are very slim so you will end up with a grey "stain" instead of this one.

Re: Washing a baseball cap........
 

The mark on the front left panel (as worn) does look like a stain or accelerated fade from something getting onto/onto the fabric. The panel that is now a different colour from the rest of the cap is almost certainly because that panel was from a different batch of fabric back in the factory - it will have looked the same originally but has faded differently over time.

Re: Washing a baseball cap........
 

That front mark still looks as though there is grease/oil in the fabric around/above the lightish brown bit. But not likely to go back to being grey even if that comes out.