There is no general consensus regarding whether a ring should have a unity element or not. Many authors work with unital rings , and other does not essentially require unity. 
If we do not assume unity to be a necessary part of ring, lets call that structure , a "**rng**" (which may or may not have unity i.e. $1$), then many pathologies do occur. 

I am hoping to get such pathologies (with at least one example) listed together on a page which I can keep as a record for future references. 

I will post my list as an answer here, to get it started, and hope others will contribute to it, and make this post valuable.