AI chatbots, built using large language models, are increasingly integrated into society and mimic the patterns of human text exchanges. While previous research has raised concerns that humans may form romantic attachment to chatbots, the range of AI-mediated interactions that people wish to create for themselves or others with chatbots remains poorly understood, particularly given the fast evolving landscape of chatbots. We provide an empirical study of Character.AI (cAI), a popular chatbot platform that enables users to design and share character-based bots, and synthesize this with an analysis of Reddit posts from cAI users. Contrary to popular narratives, we identify that users want to: (1) engage in intimate role-play with young adult, masculine-presenting characters that place users in a position of inferior power in well-defined scenarios and (2) immerse themselves in boundless, fantasy settings. We further find that users problematize both the excessive and insufficient sexualized content in such interactions which warrants novel digital-safety features.