"Too many files open" error when syncing an iPhone
This appears to be a bug with current versions of iTunes, and may be fixed in the future.

In the meantime, a workaround which sometimes works:


Disable "automatically sync this phone when connected"
Turn off music synchronization, and click "remove" when it asks you if you really want to stop synchronizing to this library
Sync the phone manually
Turn music synchronization back on, and then click "remove and sync" when it asks you what to do.
Sync the phone manually


After this point the library seems to be repaired and it starts to sync again, at least for a little while.

UPDATE: This technique stopped working for me (or rather, it failed to work; thanks, confirmation bias). However, doing a backup of my phone, and then restoring it from that backup, usually works. Eventually.