Research in Fairness, Accountability, Transparency, and Ethics (FATE)1 has established many sources and forms of algorithmic harm, in domains as diverse as health care, finance, policing, and recommendations. Much work remains to be done to mitigate the serious harms of these ystems, particularly those disproportionately affecting marginalized communities. Despite these ongoing harms, new systems are being developed and deployed, typically without strong regulatory barriers, threatening the perpetuation of the same harms and the creation of novel ones. In response, the FATE community has emphasized the importance of anticipating harms, rather than just responding to them. Anticipation of harms is especially important given the rapid pace of developments in machine learning (ML). Our work focuses on the anticipation of harms from increasingly agentic systems. Rather than providing a definition of agency as a binary property, we identify 4 key characteristics which, particularly in combination, tend to increase the agency of a given algorithmic system: underspecification, directness of impact, goal-directedness, and long-term planning. We also discuss important harms which arise from increasing agency – notably, these include systemic and/or long-range impacts, often on marginalized or unconsidered stakeholders. We emphasize that recognizing agency of algorithmic systems does not absolve or shift the human responsibility for algorithmic harms. Rather, we use the term agency to highlight the increasingly evident fact that ML systems are not fully under human control. Our work explores increasingly agentic algorithmic systems in three parts. First, we explain the notion of an increase in agency for algorithmic systems in the context of diverse perspectives on agency across disciplines. Second, we argue for the need to anticipate harms from increasingly agentic systems. Third, we discuss important harms from increasingly agentic systems and ways forward for addressing them. We conclude by reflecting on implications of our work for anticipating algorithmic harms from emerging systems.