That's a great category.   Some names that come to mind:

Willie Stargell, Dale Murphy, Carl Yastremski

I was on a plane with the Twins players going from Minneapolis to Baltimore in about 1969 to the All Star game.  There was Killebrew (husky but much shorter than I expected), Carew (he was a real sharp dresser in a suit that didn't have a wrinkle in it and he was carrying his bats in what looked like big resin bags), Jim Katt, Johnny Roseboro (he replaced Roy Campenella for the Dodgers after RC's paralyzing accident) and I found out that Roseboro and some other Dodgers were the singers in  the Chevy ad "See the USA in your Chevrolet".  His most famous incident was when Juan Marichal hit 2 Dodgers with pitches and Sandy Koufax would not retaliate to help protect his teammates.  Roseboro was incensed and when Marichal came to bat he threw the ball very close to his face when returning the pitches to Koufax.  Marichal got amped up and hit Roseboro over the head three times with his bat, opening a big gash.  Surprise - fight started.

There are so many hidden stories (such as the Chevy singing gig) in baseball that you could have a whole series of Paul Harvey's famous "..and that's the rest of the story" episodes.  Reminds me, I do sorely miss Paul Harvey.  He sure knew how to use his voice and tell a story.  Makes you realize no matter what you do in this life you are pretty much forgotten after a few years except by your family and maybe a few good friends.  Makes me laugh at the "donors" who give extravagant amounts of money to public buildings, college buildings, etc and have these donations named after them just to preserve their names in history for an extra 50-100 years.