The goal of the multi-sound source localization task is to localize sound sources from the mixture individually. While recent multi-sound source localization methods have shown improved performance, they face challenges due to their reliance on prior information about the number of objects to be separated. In this paper, to overcome this limitation, we present a novel multi-sound source localization method that can perform localization without prior knowledge of the number of sound sources. To achieve this goal, we propose an iterative object identification (IOI) module, which can recognize sound-making objects in an iterative manner. After finding the regions of sound-making objects, we devise object similarity-aware clustering (OSC) loss to guide the IOI module to effectively combine regions of the same object but also distinguish between different objects and backgrounds. It enables our method to perform accurate localization of sound-making objects without any prior knowledge. Extensive experimental results on the MUSIC and VGGSound benchmarks show the significant performance improvements of the proposed method over the existing methods for both single and multi-source. Our code is available at: https://github.com/VisualAIKHU/NoPrior_MultiSSL