There seems to be something wrong with the original question. If f(x)=-5, the answer should be -5, but I think it should be f( 1)=-5.

Solution: from f(x+2)= 1/f(x), we can get f (x+4) = f (x+2+2) =1/f (x+2) = f (x).

So f(x) is a periodic function with a period of 4.

Therefore, f (f (5)) = f (f (1)) = f (-5) = f (-1) =1/5.