Summary: In this manuscript we deal with regularity issues and the asymptotic behaviour (as \(p \rightarrow \infty \)) of solutions for elliptic free boundary problems of \(p\)-Laplacian type (\(2 \le p< \infty \)):
\[-\Delta _p u(x) + \lambda _0(x)\chi _{\{u>0\}}(x) = 0 \quad \text{in} \quad \Omega \subset{\mathbb{R}}^N, \] with a prescribed Dirichlet boundary data, where \(\lambda _0>0\) is a bounded function and \(\Omega \) is a regular domain. First, we prove the convergence as \(p\rightarrow \infty \) of any family of solutions \((u_p)_{p\ge 2}\), as well as we obtain the corresponding limit operator (in non-divergence form) ruling the limit equation, \[\begin{cases} \max \left\{ -\Delta _{\infty } u_{\infty }, \,\, -|\nabla u_{\infty }| + \chi _{\{u_{\infty }>0\}}\right\}&= 0 \text{ in }\Omega \cap \{u_{\infty } \ge 0\} \\ u_{\infty }& =F \text{ on } \partial \Omega . \end{cases} \]Next, we obtain uniqueness for solutions to this limit problem. Finally, we show that any solution to the limit operator is a limit of value functions for a specific Tug-of-War game.