Most of the times they don't ask you to show any additional documents - except your I-485 approval notice, passport, ADIT pictures and I-94.
For me and my wife, we carried the whole immigration folders that we maintaioned carefully for last six years (starting from my very first H1 Notice of Action). I guess it's Ok and safe if you carry just following documents.
1. passport
2. I-94
3. ADIT pictures
4. I-485 approval notice
5. All Advance parole (including all that expired)
6. All EAD cards (including all that expired)
7. Drivers license
8. Social sec card
9. The last tax return
10. latest paystubs
11. if you are married:
(A) marriage certificate
(B) rental agreement/mortgage/utility bills/credit card/joint-bank statements that shows both of your names
