I'm using Ungar's terminology and notations. In the open unit ball of $\mathbb{R}^n$, let $GB(A_1, \ldots, A_N; m_1, \ldots, m_N)$ be the gyrobarycenter of the points $(A_1, \ldots, A_N)$ with gyrobarycentric coordinates $(m_1, \ldots, m_N)$.

I numerically found that (for $n=2$, I have not checked for other values of $n$) $$ GB(A_1,A_2,A_3;1,1,1) = GB\bigl(GB(A_1,A_2;1,1), A_3; 2\gamma_{(-A_1)\oplus A_2}, 1\bigr) $$ and $$ GB(A_1,A_2,A_3;2,2,1) = GB\bigl(GB(A_1,A_2;1,1), A_3; 4\gamma_{(-A_1)\oplus A_2}, 1\bigr). $$ I also found that there is $p$ not depending on $A_3$ such that $$ GB(A_1,A_2,A_3;2,1,1) = GB\bigl(GB(A_1,A_2;2,1), A_3; p, 1\bigr) $$ but I didn't find the expression of $p$.

So it looks like there is an associativity property of the gyrobarycenter. What is this property? There is nothing about that in Ungar's books.