What is the purpose/function of ".ßßß" files?

Most likely these aren't actual files but the result of filesystem corruption.

• It is normally not possible to have multiple identically named files.
• Their names (ßßßßßßßß.ßßß) correspond to hexadecimal bytes E1 E1 E1… in code page 437 (which was the default MS-DOS code page, and therefore the default FAT/FAT32 code page when long file names aren't in use).
  (The character is not the Greek beta but the German lowercase sharp S. The dot isn't actually stored in the FAT, but added by the OS when reading, so it doesn't get corrupted.)
• Their sizes are close to 3 789 677 025 bytes, which is again 0xE1 E1 E1 E1 in hex.
  (That's approximately 3 700 856.469 kilobytes; Windows probably rounds up.)

All signs point to part of your filesystem's master file table being filled with the byte 0xE1, which may be caused by software (such as unplugging mid-write), but may also indicate that the flash memory itself is dying.

Software-induced corruption can often be cleaned up by using Windows' disk error checking (chkdsk). Or just reformat the drive (after copying your real files out of it).

But especially for cheaper and/or heavily-used drives, bad flash memory is very likely. Don't use this drive for important files anymore.