Piecewise function

f(1)=-1

$y=-x^2$, and $y=1$

$1=-x^2$, solve for x, $x=-1$, check it fits piecewise, and it does.

therefore $f(1)=-1$

You can also graph the equations $y=-x^2$ with $x \geq0$  and $y=x+8$ with $x<0$, go to where $y=1$ and see what $x$ is.

f(1)  =   - (1)^2  =  -1

f (f(1) )  =  f (-1)  = -1 + 8  =  7

f (f (f (1) ) )  =  f ( 7)  = - (7)^2  = -49

 f ( f ( f( f(1) ) ) )   =  f (-49)  = -49 + 8  = -41

f ( f ( f ( f ( f ( 1) ) ) ) )  =  f (-41)  =  - (-41)^2  = -1681