Theoretical models of artificial general intelligence, such as AIXI , typically consider an intelligent agent to have unlimited computational resources, allowing it to keep a perfect memory of its entire interaction history with its environment. In the real world, an agent’s memory is part of the environment, which means that the latter can modify the former. This paper develops a theoretical framework for examining the implications of such real-world memory on universal intelligent agents. Within this framework we are able to show, for example, that in certain environments optimality can be achieved only with truly stochastic behaviors, and that guarantees about the trustworthiness of memories are difficult to obtain even with infinite computational power. To describe the probability of an agent’s memory state, we propose an adaptation of the universal prior for the passive and the active case.