If $p$ is a bijection between a set $M$ and $M^2$, we say that it is a pairing function with no cycles if for any term $t(x_1,..., x_n)$ formed with $p$, and that is not a variable, we have, for every $a_1,\ldots, a_n \in M$, $t(a1,\ldots, a_n) \neq a_1$. The theory of pairing function with no cycles is the simplest example of a stable theory that is not a limit of super stable theories. It has been studied by several authors and, in particular, it has been shown that it is complete and admits quantifier elimintation in a natural language. Grothendieck rings of a structure have been introduced in Model Theory in 2000. Its construction relies on indentifying definable sets that are in definable bijection and generalizes the definition of Grothendieck ring already known in algebraic geometry. We will first give a brief survey of this model theoretic notion. Then we'll compute the Grothendieck ring of pairing function with no cycles, and show that it is isomorphic to $\mathbb{Z}[x]/(X-X^2)$.