Is there any substance that's a 4-4-4 on the NFPA diamond?
Answering my own question based on the comments, tert-butyl-hydroperoxide is at least one such chemical. As stated on this MSDS from a government website, it's a 4-4-4, with additional special warning of being a strong oxidizer. The only thing that it does not do that could make the 704 diamond any worse is react strongly with water. It is in fact water soluble, though marginally, preferring to float on top (and therefore traditional water-based fire suppression is ineffective, but foam/CO2 will work).

If anyone else can find a chemical that, in a form that is used in the lab or industrially, is a 4-4-4 that is a strong oxidizer and reacts strongly with water, that's pretty much "as bad as it gets" and they'll get the check.