I came to game design from an experimental science background. In the biochemistry lab, I prove x exists by developing an assay for the activity of x. If I can’t test for x, I can’t measure x. I can’t say, “There should be a lot of x in this test tube.” I can’t say, this test tube looks like a lot of x to me.” I can’t even say, “This test tube was prepared the same way I always prepare, so it should have the same amount of x as always.”
If I want to talk about x, I need an assay for x. If I want to optimize x, I need that assay for x. How do I know I have more or less, if I can’t measure it? Should I add this or subtract that to increase the concentration of x? If I have an assay, it’s a straightforward question. If no assay, it’s impossible.
So, what is an assay for fun? I don’t think there is one. This is the point of the OP’s statement, I think. This is why people write books about fun. Once we start applying an experimental science approach to game design, that is, once we start doing User Experience Research, then we all come to the same conclusion: there is no assay for fun.
Therefore, fun becomes a concept as magical as the concept “life”. (Long fun discussion down that path….. )
For folks who came to game design from a non experimental path, this realization is evidence of their graduation into scientific thought. It’s a fun place to be, welcome!!
Now, what meaningful things can we assay for? Personally, I assay for a willingness to press the “Start next level” button. If start next level is pressed, I conclude the just played level was well designed. Of course, the conditions under which I conduct this assay make a difference. When I test my games at an expo or online, where players feel free to come and go freely, I get fewer levels played than when I test my games in say, a classroom. In a classroom, students of all ages are likely to play a game longer. Significantly longer. (long informative discussion about learning game design and in particular, how learning games could easily be more complex than they typically are because their audience starts out more invested…. Oh, and learning game devs can save money by allowing their tutorials to be more like old school game booklets rather than the finely crafted tutorials necessary to capture say, the app store audience….)
Here are a lot of assays that one could use to measure fun related factors.
How many levels do players play?
Do players come back and play more
Do players buy the game?
Do players recommend the game to their friends?
Do players finish the game?
So players feel like they are learning in the game?
Do players remember what they did in the game?
Do players feel competent while they play?