Given $n\in \mathbb N$, is there a free module with a basis of size $m$, $\forall m\geq n$?
The example you gave is the canonical example of a ring without Invariant Basis Number, and aside from that, I only know about one other family of rings produced with Leavitt path algebras.
I think I'm remembering right that their key feature that set them apart from the example you gave was that they could produce this $R^n\cong R^m$ behavior for prescribed $m,n\in \Bbb N$.
Check them out in Abrams and Anh's paper!
Added: Hmph, I guess I forgot that link wasn't accessible to everyone! Anyhow, I found a slideshow Gene made that will also do the trick.