Let $\boldsymbol{V}_{1},\dots,\boldsymbol{V}_{n}\in\mathbb{R}^{d\times m}$ be $n$ “tall” matrices (where $d\ge m$) with orthonormal columns.
And let $\boldsymbol{P}_{1},\dots,\boldsymbol{P}_{n}\in\mathbb{R}^{d\times d}$ be the orthogonal projection matrices defined as $\boldsymbol{P}_{i}=\boldsymbol{V}_{i}\boldsymbol{V}_{i}^{\top}$.
Finally, let $\boldsymbol{T}$ be an operator defined the concatenation of the projection matrices $\boldsymbol{T}=\boldsymbol{P}_{n}\cdots\boldsymbol{P}_{1}$.
Question: does there exists a sequence $\boldsymbol{V}_{1},\dots,\boldsymbol{V}_{n}$ such that the concatenation operator $\boldsymbol{T}$ is nilpotent with index $k\ge3$? That is,
$$\boldsymbol{T}^{k-1}=\left(\boldsymbol{P}_{n}\cdots\boldsymbol{P}_{1}\right)^{k-1}\neq\boldsymbol{0}_{d\times d},\,\,\,\,\,\,\,\boldsymbol{T}^{k}=\left(\boldsymbol{P}_{n}\cdots\boldsymbol{P}_{1}\right)^{k}=\boldsymbol{0}_{d\times d}$$
Special case (example): when $n=3$ and $d=2$, we can choose projections such that $\boldsymbol{T}$ is a nilpotent operator with an index of $k=2$.
Choose $\boldsymbol{v}_{1}=\left[0,1\right]^{\top},
\boldsymbol{v}_{2}=\frac{1}{\sqrt{2}}\left[1,1\right]^{\top},\boldsymbol{v}_{3}=\left[1,0\right]^{\top}$.
Then, the projection matrices are
$$\boldsymbol{P}_{1}=\boldsymbol{v}_{1}\boldsymbol{v}_{1}^{\top}=\left[\begin{array}{cc}
0 & 0\\
0 & 1
\end{array}\right],\,\,\,\boldsymbol{P}_{2}=\frac{1}{2}\left[\begin{array}{cc}
1 & 1\\
1 & 1
\end{array}\right],\,\,\,\boldsymbol{P}_{3}=\left[\begin{array}{cc}
1 & 0\\
0 & 0
\end{array}\right]$$
And the concatenation of these matrices is
$$\boldsymbol{T}=\boldsymbol{P}_{3}\boldsymbol{P}_{2}\boldsymbol{P}_{1}=\frac{1}{2}\left[\begin{array}{cc}
0 & 1\\
0 & 0
\end{array}\right]$$
which is a nilpotent matrix with an index of $k=2$ (i.e., $\boldsymbol{T}^{2}=\boldsymbol{0}_{d\times d}$).
Best Answer
I think your example is easily generalisable for any index. For example, let $$ Q_1=P_1\oplus(1), Q_2=P_2\oplus(1), Q_3=P_3\oplus(1)=(1)\oplus P_1, Q_4=(1)\oplus P_2, Q_5=(1)\oplus P_3. $$ Then $Q_5Q_4Q_3Q_2Q_1$ is nilpotent of index 3, and I think that a similar construction with $2n-1$ matrices of size $n$ gives nilpotence of index $n-1$.