George Andrews and Bruce Berndt have written five books about Ramanujan's lost notebook, which was actually not a notebook but a pile of notes Andrews found in 1976 in a box at the Wren Library at Trinity College, Cambridge. In 2019 Berndt wrote about the last unproved identity in the lost notebook:
Following Timothy Chow's advice, I consulted Berndt and asked him if there were any remaining formulas of Ramanujan that have neither been proved nor disproved. He said no:
To the best of my knowledge, there are no claims or conjectures remaining. There are some statements to which we have not been able to attach meaning.
I checked to make sure that this applies to all of Ramanujan's output, not just the lost notebook, and he said yes.
EDIT: However, only on December 21st, 2021 did Örs Rebák submit this paper to the arXiv:
in which he completed an incomplete formula in Ramanujan's lost notebook, and proved it. So there may still be gems left to polish.