In the spirit of the automorphism tower problem, I've been thinking about "$K_0$ towers." Since $K_0$ may be imbued with a ring structure by the tensor product, it makes sense to ask what that ring's $K_0$ is. There are two fixed points that are immediate to me: $0$ and $\mathbb{Z}$.
As an example, consider the case $L_\mathbb{C}(1,3)$, the Leavitt algebra with module type $(1,3)$. Then we get the following $K_0$ tower: $L_\mathbb{C}(1,3), \mathbb{F}_2, \mathbb{Z},\mathbb{Z},\mathbb{Z},\dots$
In fact, all the examples I've tried seem to terminate in $0$ or $\mathbb{Z}$ rather quickly so it would be interesting to see other behaviors. I would at least like to know if there are other fixed points or perhaps even cycles, but a complete characterization of behaviors would be ideal.
