This drives me absolutely crazy and I'm hearing it often from intelligent people.

Example:
"Greg and I's relationship."

Should it not be the following:
"Greg's and my relationship"

Am I wrong?

"Greg's and my relationship" is preferable to "Greg and I's relationship" (which is wrong).

"Mine and Greg's relationship" is the best way to say it.

Here's a quick lesson joint ownership with apostrophes. (It's at the very bottom of the page.)