This transcript doesn't mention it specifically but the most accurate prediction they're referring to is likely the anonalous magnetic dipole moment of an electron [1] where theory matches experimental value to at least 10 significant digits. The article talks about 12 decimal places so maybe they're referring to somethign else?

At the other end of the spectrum is the s-called vacuum catastrophe [2] where the predicted and actual values for the energy density of a vacuum diverge by as many as 120 orders of magnitude.

As for the incompleteness of QFT, this is well beyond my knowledge. I really wish this were an article instead of a transcript though. Transcripts are such poor means of conveying information.

[1]: https://en.wikipedia.org/wiki/Anomalous_magnetic_dipole_mome...

[2]: https://en.wikipedia.org/wiki/Cosmological_constant_problem