What happens when a creature with persist dies and already has a -1/-1 counter on it?

Persist reads: 'When this creature dies, if it had no -1/-1 counters on it, return it to the battlefield under its owner's control with a -1/-1 counter on it.' The key phrase is 'if it had no -1/-1 counters on it' — this is a condition checked at the moment the creature died (i.e., just before it left the battlefield, using last-known information). If even one -1/-1 counter was on the creature when it died, the persist ability does not trigger at all.

This is defined in CR 702.79a, which states the condition must be met for the triggered ability to function. CR 603.4 further clarifies that a triggered ability that has an 'if' condition in its trigger clause simply does not trigger when that condition isn't met.

The last-known information rule (CR 112.7a, CR 603.6d) means the game looks at the creature's state as it last existed on the battlefield to evaluate whether the condition was satisfied — so counters present at the moment of death are what matter.

Concrete example: You control a Murderous Redcap (which has persist). An opponent uses Puncture Blast to put a -1/-1 counter on it, and the counter reduces it to 0 toughness. Murderous Redcap dies, but because it already had a -1/-1 counter when it died, persist does not trigger and it stays in the graveyard.

This interaction is also why combining persist with undying or with ways to remove -1/-1 counters (like Melira, Sylvok Outcast) creates powerful loops — the counter condition is what limits the recursion.