The constraints you place on the project, i.e., no analog, no microcontroller, severely limit what you will be able to do. From a medical standpoint, most things begin as analog, such a blood pressure, blood glucose, temperature, etc. The lack of a microcontroller will limit the amount of logic/timing yo can do.
You might want to consider something that measures human response and use simple logic to compare different subjects. There are some very interesting experiments that compare human to animal (monkey) in ability to distinguish numbers and friend or foe. (Example: Flash a screen with 8 dots, then one with only 7. Ask which had more dots.) Here are some references in the popular media to that type of study. If it is an area you want to pursue, let us know and I will find some better articles.
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Monkeys Keep Turning Out to Be Smarter than People Think They Are - Softpedia
The studies are often done using monitors to present the challenges. That would probably not fit within the constraints you place. However, you could use flash cards to present the challenges. Also, I assume you neither have nor want access to monkeys. For a project, you could apply the same principle of testing to a few friends -- maybe compare female to male response, old to young, hunters to non-hunters, gamers to non-gamers, math majors to history majors, etc.
Finally, the electronic part could use logic to determine response speed and correctness of your subjects. That is, something similar to a game or quiz show buzzer. If you search on that term you will get many hits. Here are just the first three I got so you can see what I mean:
Game Circuits
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Quiz Show Buzzer System using Staples Easy Button
John
Edit: Additional thought
Here's another idea: Have a series of switches to enter a truth table. Then have your circuit determine what type of gate was entered. The answer for each gate type could be shown with an LED.