Hello! I feel like this is a question that I swear I’ve seen you answer before a while back, but I can’t remember what you might have said and searching your blog isn’t pulling up what I thought I remembered or much information. But my question is, what examples might there be of prayers from Old Norse times (or something that could be passed off as a prayer, with a bit of creativity)? I feel like the answer was “not much of anything like that survives to the present day”, but I can’t remember if/what you’ve answered before. Thank you and sorry for the repeat question if you’ve answered something similar!
Some years ago, Ross Downing put together a book of heathen prayers compiling everything he could find from pre-modern sources. His website has been down for a while and I can't seem to find a way to access the book as a PDF, but there's a reel of maybe the whole thing on his instagram. I think it's a stretch to call some of what's included "prayer" (and he fudges some translation like on the Sparlösa runestone) but that does still seem to fit your criteria.
Interesting article by John Lindow about prayers to Thor (generally in the form of recounting times when he kicked ass): https://www.jstor.org/stable/40918940
I think some of Freyja's opening stanzas in Hyndluljóð are arguably prayers or at least prayer-like, as well as the final line of the poem: Bið ek Óttari öll goð duga 'I ask that all the gods aid Óttarr' (biðja is actually also the word used for 'pray' though in other contexts it just means 'ask' or 'request').
I don't know if this is the sort of thing you're looking for but there are examples of poems and formulae in later folklore that you might be able to frame as prayers depending on how you define it, though they were not likely conceived as such in their own time. An example that comes to mind is Galdra-Brandur's invocation of Gylfi (here, presumably a being related to cold like a hrímþurs, rather than a human king) to wipe out swarms of biting flies:
Gylfi hæða galhvassan gefi vind á landnorðan með óveðri magnaðan mývarginn svo drepi hann.
(roughly) 'May Gylfi give harsh highland wind in the north with magnified storm-weather so may he kill the mývargar (biting flies).'
But that's way later than Old Norse times and not a reliable witness to a person actually speaking (Galdra-Brandur was a real guy but known mostly through tall tales).
[Edit] I didn't specifically mention because it's so well-known already and because it's included in Ross's book, but on second thought the Heilir æsir/hailar ásynjur part of Sigrdrífumál is probably too important not to at least mention it.
