QR decomposition properties

it is just out of curiosity, lets say we take QR decomposition A = QR and take QR again and again infinitely, A= Q(Q(Q....))R, would the R be converged for any matrices?. I am looking for its analysis.

From the looks of it, you're looking at writing $A=QR$ and then QR decomposing $Q$ etc. This doesn't do anything very interesting because the QR decomposition of an orthogonal matrix $Q$ is just $QI$ where $I$ is the identity matrix of size equal to the number of columns of $Q$.

The flip side, where you write $A=QR$ and then QR decompose $R$, is also not very interesting because a QR decomposition of an upper triangular matrix $R$ with nonnegative diagonal elements is just $I R$, where $I$ is the identity matrix with size equal to the number of rows of $R$.

There is an algorithm like this called the QR algorithm in which you iteratively define $A_{k+1}=R(A_k) Q(A_k)$ where $R(A)$ and $Q(A)$ are QR factors of $A$. (Note the change in the order of $Q$ and $R$.) It can be used (usually with some modification) to find eigenvalues and eigenvectors.