$Q$ is a $n_Q \times m_Q$ matrix and has known rank $r_Q$
$P$ is a $n_P \times m_P$ matrix and has known rank $r_P$
Let $q_k$ and $p_k$ be the k-th kolumn of $Q$ and $P$ resp.
My goal
Find an expression of rank $R$ in function of properties of Q and P.
The matrix R
\begin{equation} R = \left[ \begin{array}{@{}c|c|c@{}} Q & Q & Q & \cdots\\ p_1\ \ p_1 \cdots & p_2\ \ p_2 \cdots & p_3\ \ p_3 \cdots \end{array} \right] \end{equation}
\begin{equation} \hat{R} = \left[ \begin{array}{@{}c|c@{}} q_1\ \ q_1 \cdots & q_2\ \ q_2 \cdots & q_3\ \ q_3 \cdots \\ P & P & P & \cdots \end{array} \right] \end{equation}
It can be easily seen that $\text{rank}(R) = \text{rank}(\hat{R})$ because it is just a rearrangement of the columns.
Results so far
Based on intuition and experimental results I believe that there exists an upper bound: $\text{rank}(R) \le \text{rank}(Q) + \text{rank}(P)$
I have no proof of this though.
Update
Based on the suggestions the problem is reduced to the following problem: I substracted the first column from every other column.
\begin{equation} \text{rank}(R) = \text{rank}( \left[ \begin{array}{@{}c|c|c@{}} Q \\ p_1\ \ p_1 \cdots \end{array} \right]) + \text{rank}(R) - 1 \end{equation}