Considerable attention in reliability literature has been given to studying various repair models. These models are often described under the assumption that the repair being conducted is at least minimal. In fact, in recent literature, a multi-attempt minimal repair model has been described, where an attempt to repair a system after failure can be unsuccessful and, therefore, should be repeated until a successful minimal repair. This assumption of minimal repairs is not unrealistic. However, it can still be too restrictive in that it does not capture the adverse effects of the previous repair attempts, external and internal shocks to the system, insufficient quality of a successful repair attempt, etc. Therefore, in this research, we use the generalized Polya process to define and derive useful properties for a more general multiple-repair-attempt process. This has the minimal repair model as a special case. The corresponding age replacement policy for the general multiple-repair-attempt model is defined, and the optimal solutions are obtained. Detailed numerical illustrations support our findings.