GnuTLS doesn't actually log very much at log level 1.

It currently logs things at the following levels:

level 3: asserts
level 4: handshake information
level 5: record-layer information
level 9: session keys, IVs, and internal secrets
level 10: raw network read and buffering information

level 5 seems like the right layer for "verbose", without leaking too
much sensitive stuff.