Games centered around a "groundhog day" loop - an IFDB Pollby Merk
Two that come to mind, which I haven't played in years and may be remembering wrong, are Moebius and All Things Devours. Games with fail states, by their nature, fit the bill from a mechanical level, but I'm curious about games where that's actually the focus of the plot. One-move games probably fit the bill too, but I'm also looking for examples of games that are maybe a little more complex, and where the repeating loop is used in clever ways.
(This is for an ongoing discussion topic for my podcast, where we've been talking about Groundhog Day plots in games. There are a few examples in video games, but I got to thinking that the best use has probably been in interactive fiction.)
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Time loops are perhaps most prominent in choice-based games, where they're a foundational element: time loops first appear in the very first CYOA, Sugarcane Island. (Some discussion at https://heterogenoustasks.wordpress.com/2011/08/10/cyoa-structures-more-packard/ .) There are too many examples to cover all of them - time travel is evidently just something that CYOA naturally suggests.
You might want to read part of Jeremy Douglass's PhD: from page 277 to roughly the end, he talks about Aisle, Shrapnel and Rematch and what time loops mean for IF as a form. Haven't gotten to that part though, but it's super interesting so far so I'd bet there's good stuff in there :)