There's a few things to navigate and untangle in your question, but the short answer is YES, there are martial arts out there that do this.
Mostly it comes down to instructor rather than specific style, although obviously certain styles tend to be grouped around testing, you can find instructors who do not participate in that manner.
Holding Back Information
Competition styles tend not to hold back information that much. People exit out of competing age and then go on to train other people. These teachers have a vested interest in seeing their students win and earn titles and trophies as a testament to their own teaching skill. So you tend not to have a lot of holding back there. So it's actually the opposite of what you're suggesting in your question.
The only restriction is that some styles are primarily sport over combative, in which case you may miss out on some combative elements solely due to the focus of competitive training. Whether you consider tournaments to be an "exam" or not, that's definitely one arena where you don't have a lot of that kind of problem of withheld secret information.
Traditional martial arts built on the instructor-disciple foundation tend to suffer the most from withheld information. There's a variety of possible reasons - favoritism towards blood relatives/descendents, religious requirements, not being given permission to teach by someone higher in authority than the teacher, fear of being "out done" by the studnets, etc. This has hurt a lot of martial arts over time, and many people in these lineages are often at odds with the problems of fragmented and partial information.
Now this is a different issue.
If, by exams, you mean, "repeat these factoids about this art", "perform these 3 movement sets", etc. there's many martial arts that do not do this type of exam. These exams and levels of measurement are relatively recent in the last few generations and serve as a method of standardization, which has both strengths and weaknesses for promoting an art.
If, by exams, you mean, "Does anyone judge or require to see how well I'm progressing?" that's a key part of pretty much learning ANYTHING. Your instructor should be watching what you are doing and giving you feedback so you can fix errors and improve strengths. Your fellow students should be giving you advice and helping you grow as well. You should be engaging in self observation during your training and seeing what you can work on and giving yourself credit for where you improve (sometimes, it feels invisible).
Sparring, high intensity drills, or other key parts of your training can serve as informal "exams". Many schools may have no exam or test unless you decide to become an instructor, in which case, that's the only test you need to worry about.
"What Style Should I Take?"
As I said, it depends on the instructor and school you find. Let them know what you're looking for (self defense, fitness, etc.) and the fact that you are not interested in doing testing, or having to unlock "secrets". Just as much as you're looking for a good fit, many teachers and schools are doing the same and it's a great way to sort things out from there.