Exploration of unknown, unstructured environments, such as in search and rescue, cave exploration, and planetary missions,presents significant challenges due to their unpredictable nature. This unpredictability can lead to inefficient path planning and potential mission failures. We propose a multi-objective risk assessment method for exploration planning in such unconstrained environments. Our approach dynamically adjusts the weight of various risk factors to prevent the robot from undertaking lethal actions too early in the mission. By gradually increasing the allowable risk as the mission progresses, our method enables more efficient exploration. We evaluate risk based on environmental terrain properties, including elevation, slope, roughness, and traversability, and account for factors like battery life, mission duration, and travel distance. Our method is validated through experiments in various subterranean simulated cave environments. The results demonstrate that our approach ensures consistent exploration without incurring lethal actions, while introducing minimal computational overhead to the planning process.