Ancient Coin (Possibly Roman?)

Going solely off of the bust, I would say late 1st to late 2nd centuries.

Well maybe you can read some of the lettering around the edge?

Even if you just have to guess like : "18 letters, ? D or O ?..." it would be a start. If you can make out any greek letters like pi or omega or lambda that would also provide a lead. You can also try taking obliqued pictures from different angles to highlight the lettering.

Some collectors are good at IDing the portrait and can say "That looks like emperor so-and-so" but I am not that experienced.

So I'd say ID the lettering would be your best bet at definitive ID. Even if one of us were to suggest a match, one of the first things you'd do is check to see if the lettering matches.

This weekend I tried checking the coin under a magnifyer & photographing under different lighting, without any effect. I could make out none of the lettering as it was worn away. I honestly thought that it would remain as unidentified.
That all changed with the post by t0rress! Thank you so much for being able to identify the bust as Trajan. I don't think it will get a definite ID between the two coin images as the figure is so worn as well.
Thank you again t0rress, you have made my weekend.