I wish he were rich implies the present. (today)
I wished he were rich implies the past. (for example, in 1966)
"I wish he had been rich" implies that he wasn't rich. (for example, in 1966)
If you say "I wished he had been rich", with "had been", as you can see, still there, it will imply that, say, I wished (in 1967) he had been rich (in 1966).
Student or Learner