jmc

We want the size of the set $f^{-1}(f^{-1}(f^{-1}(f^{-1}(3)))).$ Note that $f(x)=(x-1{)}^2-1=3$ has two solutions: $x=3$ and $x=-1$, and that the fixed points $f(x)=x$ are $x=3$ and $x=0$. Therefore, the number of real solutions is equal to the number of distinct real numbers $c$ such that $c=3$, $c=-1$, $f(c)=-1$ or $f(f(c))=-1$, or $f(f(f(c)))=-1$.

The equation $f(x)=-1$ has exactly one root $x=1$. Thus, the last three equations are equivalent to $c=1,f(c)=1$, and $f(f(c))=1$. $f(c)=1$ has two solutions, $c=1\pm \sqrt{2}$, and for each of these two values $c$ there are two preimages. It follows that the answer is $1+1+1+2+4=\boxed{9}$.