7 hours ago I asked a question involving a rare time signature (Why does 1/1 time signature exist if you are curious), but then my curiosity had me thinking about X/32, X/64, and absurd time signatures like that and I wonder, are there any songs that use something like that? The only thing time I found a time signature like that is some joke piece on musescore on 57/64 for which I don't remember the link.
Since this is an existence question, the answer is "yes", and here's an example:
Telemann uses 3/32 in the second movement of his Gulliver Suite, "Lilliputsche Chaconne" ("Chaconne of the Lilliputians").
Benjamin Pesetsky, on his website, gives an outstanding explanation of why Telemann did this:
Telemann writes the Chaconne of the Lilliputians in the very small time signature of 3/32, a visual joke for the violinists, who have to read miniscule 64th and 128th notes.
Further, the Lilliputians being famously small, and the Brobdingnagians famously large:
The Brobdingnagians are written in the equally eccentric time signature of 24/1, putting their music into enormous whole notes.
Note in this video from a recital at UCLA how the violinists communicate the idea of Lilliputians to the audience.
Strictly speaking, time signatures like this aren't necessary; they could easily be rewritten using a more accustomed notation. However, in addition to humor value, then can also communicate proportions to surrounding music, or they can be suggestive of how to interpret the music. "Small" notes tend to connote "faster" music, but they also can be easier to read if a piece is especially slow.
Wikipedia has a list of pieces with unusual time signatures that includes several in X/32 and X/64.