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.
Have you ever taken a personality quiz/test that helped you have valuable insights? If so, what were the tests and how were they useful?
The only useful ones I've found all yielded the same type of insight. They showed me where I stand relative to others, which is can be genuinely useful since representative samples of large populations can be hard to come by. This includes IQ tests and tests for mental disorders (in my experience, people are usually aware that they are, for example, smarter than the average (although the Dunning-Kruger effect might complicate this) or have some intrusive thoughts and compulsive rituals, but might be surprised to find that they are three standard deviations above the norm or that their symptoms are sufficiently severe to be considered OCD).
No remotely reliable (as in, not astrology) test I have ever seen has revealed genuinely surprising information for a moderately self-aware person, outside of ranking. Furthermore, they rarely gather personality data in a remotely subtle or non-transparent way ("do you like spending lots of time with large groups of people?" "yes..." "surprise, you're an extrovert!"), and thus seem super susceptible to test-takers' attempts to confirm a desired identity.
An example of a more interesting/subtle way to potentially conduct a personality test would be to use question like OKTrends' "do you like beer?" which clusters strongly with "do you have sex on the first date," and, potentially, sexual openness. Such results might be harder for manipulate (consciously or unconsciously) and could assist with deeper self-awareness.
Edited because the first link was broken.