When both players pass priority in succession, what happens to the pending chain?

The newest item on the chain resolves — and only that one item.

When both players pass in sequence without adding anything, you move to Step 4: Resolve, and the newest Finalized Chain Item resolves in full (Rules 339.1, 340.1). The chain is LIFO, so this is the top item, not the whole chain.

After that single resolution, all players get priority again and can respond before the next item resolves (FAQ #7640, Rule 340.4). If nothing is added and everyone passes again, the next item resolves — repeat until the chain is empty, at which point play returns to an Open State and (in a Showdown) Focus passes to the next player in turn order.