As far as I know, the parameters from quantified psychology that seem to have the highest descriptive value by far are g-heavy IQ and OCEAN. I do not know how stuff like EQ or different types of mental disorder tests fit into that framework, but here are some forms that psychologists can use. If there is an anonymous test, as ChristianKI suggests, I'd be willing to take it.

I wonder what the value of extensive psychological testing (IQ, OCEAN, disorders) is relative to physical tests (weight, height, blood pressure, the usual preventative measures) in terms of costs and benefits. Quite a number of people are not self-aware enough to spot the first signs of disorders themselves and to take appropriate action. Such testing, assuming high predictive value, could help with that.