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Suppose there is a creature with persist on the battlefield, for example Safehold Elite. The creature loses persist somehow, for example because a player casts Turn to Frog on it. The creature then dies, for example due to combat damage. I know that the creature card will have persist again once it's in the graveyard. Does the persist trigger go off, or does the creature simply die?

No, persist only triggers when it was on the creature at the moment it died.

Normally, triggered abilities only work when they exist by the time the relevant event has happened. However, in the example case of a creature with active persist, by the time that creature has hit the graveyard, persist is no longer active because it's a creature ability that only works on the battlefield. That would mean that abilities such as persist would never work.

Therefore, a zone-change trigger ability such as persist has to look back in time to work properly. Conversely, if persist is not on the creature when it would look back in time, it won't trigger. It doesn't matter if the new object in the graveyard has persist.

603.10a Some zone-change triggers look back in time. These are leaves-the-battlefield abilities, abilities that trigger when a card leaves a graveyard, and abilities that trigger when an object that all players can see is put into a hand or library.

702.78a Persist is a triggered ability. “Persist” means “When this permanent is put into a graveyard from the battlefield, 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.”

Key here is "triggered ability", which happen as the event occurs.

603.2. Whenever a game event or game state matches a triggered ability’s trigger event, that ability automatically triggers.

At the time the event happens, the permanent doesn't have Persist, so nothing triggers. It's only after the event concludes that Persist returns to the card.