The Markov quiver is the quiver on $[3]$ where with arrows $1 \to 2$, $2 \to 3$, and $3 \to 1$ all of multiplicity $2$. It is so named because it the cluster variables it generates can be used to construct the solutions to the Markov equation (see 3.4 of https://arxiv.org/abs/1608.05735).

For this quiver, mutating at any vertex simply reverses all the arrows so restricted mutation sequences are just the usual non-restricted mutation sequences. Since this quiver is not mutation-equivalent to a Dynkin ADE diagram, it generates infinitely-many cluster variables.