Can you combine sentence structures to denote an extremely short duration? So I learned recently that you can do two things in Chinese


repeat the verb (such as in 看看)
append 一下 to the verb to denote a shorter duration (in this case,
look for a moment).


I was just wondering, can you use both at once? For example, could I say "你看看一下我的书,好吗？" to, in this case, ask someone to look at my book for an extremely short period of time?
Short answer: no.  

As you explained, 看看 and 看一下 have the same meaning, so use both of them does not make sense.  

If you want to emphasize it's a very little bit time, you can say:
你看看我的书,好吗？就(只)看一下