I think that the  part $(a)$ of  proposition $2.4$ of [this paper ](https://arxiv.org/pdf/1506.08962.pdf)  shows that for $n$ sufficiently  large one  can construct  a $n \times n$ unitary  matrix $U=-I_{2}\oplus U'$,  which can  be  decomposed as the  product of three positive  matrices. For  such  $U$ we have $\parallel U-I\parallel=2$. In fact one  can take a unitary  matrix $U'$  with  $Det \;U'=1$ such that $0$ lies in the interior of  the  convex hull of the eigenvalues of $U$. This convex hull is equal to $W(U)$, the numerical range of $U$.

In fact  if $U=ABC$  for  positive  $A,B,C$ then $UC^{-1}=AB$  so $C^{-1}=|AB|$ is the positive  factor of $U $ in its   polar  decomposition. Of  course such  $U$ has  $-1$ as an eigenvalue.

An explicit  example is the  following 

$$\begin{pmatrix}-1& 0&0&0\\0&-1&0&0\\0&0&\sqrt{3}/2&-1/2\\0&0& 1/2&\sqrt{3}/2 \end{pmatrix} $$
According to the  aboved  linked paper this  matrix  is  a  product of  three positive  matrices. This matrix, with  distance $2$ from  the identity matrix $I_{4}$, is  the  unitary  factor  of  polar  decomposition of $AB$ for two  positive  matrices $A,B$. So  according to the notation $C_{n}$ in the  answer  by Loup  Blanc, we  have $C_{n}=2, \; \forall n\geq 4$.