Does Anyone Know What This Is?

Yes... I was so lucky... a friend of a friend mentioned this old man with a stash of old coins. Of course I had to sit through hours of talking to convince him they were going to a good home! 50 years ago I noticed a change in pennies, when wheat pennies were still in circulation and I would hide those strange pennies, like a secret treasure. Of course at 6 years old I thought I was the only one that knew or cared about those strange coins. I've collected many more coins since then, half still not identified but I have time &

Wow...a button.... you could be onto something here. I found maybe something similar but it won't let me post the image... augh.

Could be a button, but I think that normally the broken off bits on the back would be closer to the center. How about an old pin instead? The cross on the other side is similar to a red cross pin, perhaps a bit like this early 1900s Boy Scout volunteer pin (but with the enamel all missing):



Interestingly, both seem to have a textured surface on the inside of the cross (maybe to provide a rough surface for the enamel to grab?)