Recent reports on generative AI chatbot use raise concerns about its addictive potential. An in-depth understanding is imperative to minimize risks, yet AI chatbot addiction remains poorly understood. This study examines how to characterize AI chatbot addiction--why users become addicted, the symptoms commonly reported, and the distinct types it comprises. We conducted a thematic analysis of Reddit entries (n=334) across 14 subreddits where users narrated their experiences with addictive AI chatbot use, followed by an exploratory data analysis. We found: (1) users' dependence tied to the "AI Genie" phenomenon--users can get exactly anything they want with minimal effort--and marked by symptoms that align with addiction literature, (2) three distinct addiction types: Escapist Roleplay, Pseudosocial Companion, and Epistemic Rabbit Hole, (3) sexual content involved in multiple cases, and (4) recovery strategies' perceived helpfulness differ between addiction types. Our work lays empirical groundwork to inform future strategies for prevention, diagnosis, and intervention.