Does Tainted Pact exile cards until you choose to stop or until you hit a duplicate card name?

Tainted Pact reads: exile cards from the top of your library one at a time until you choose to stop or until you exile a card with the same name as a card already exiled this way. You may put the last exiled card into your hand. All other exiled cards remain exiled.

This means you have two stopping conditions. First, you may voluntarily stop at any time after exiling at least one card, and then you may take that last card into your hand. Second, if you exile a card whose name matches any card already exiled during this resolution, the spell forces you to stop immediately — and crucially, you do NOT get to put that duplicate card into your hand; it stays exiled along with all the others.

This is why Tainted Pact is most powerful in decks with no duplicate card names (often called 'singleton' or similar constructions, including Commander with a curated list). If every card in your library has a unique name, you can theoretically exile your entire library one card at a time, choosing to stop and pocket whichever card you want.

Under the Comprehensive Rules, spell instructions are followed in order (CR 608.2c), and 'until' clauses create a termination condition that ends the process immediately when met (CR 608.2). The word 'or' means either condition ending the loop is sufficient.

Example: You cast Tainted Pact and exile Island, then Forest, then another Island. The second Island shares a name with the first Island already exiled — the effect immediately stops, you cannot take the duplicate Island, and all three cards remain exiled permanently.