Kierkegaard’s “double reflection” is not hard to understand. If I like chocolate, that’s an immediate response, and can occur without reflection. If I think “I like chocolate but if I eat too much of it it will make me sick” that’s reflection. Instead of just thinking about the chocolate I’m thinking about my own response to chocolate, as if I am looking at a reflection of myself in the mirror and judging it. The double reflection is when I think “I worry about food too much — it’s making me hard to live with”. That’s forming a judgment about my own reflecting — it’s like looking at myself looking at a reflection of myself in a mirror, in another mirror. Could there be triple reflection? Sure. Here is an example “My double reflection is just an excuse I use to not deal with my chocolate addiction.”
And yet, it seems to me reflections don’t in practice advance infinitely. At some point there might be nothing to be gained by further reflection. I’m not sure if that’s (what I just wrote) an example of quaternary reflection or just plain old double reflection. And I’m not sure if it matters!