Structural extraction of events within discourse is critical since it avails a deeper understanding of communication patterns and behavior trends. Event argument extraction (EAE), at the core of event-centric understanding, is the task of identifying role-specific text spans (i.e., arguments) for a given event. Document-level EAE (DocEAE) focuses on arguments that are scattered across an entire document. In this work, we explore the capabilities of open source Large Language Models (LLMs), i.e., Flan-UL2, for the DocEAE task. To this end, we propose ULTRA, a hierarchical framework that extracts event arguments more cost-effectively -- the method needs as few as 50 annotations and doesn't require hitting costly API endpoints. Further, it alleviates the positional bias issue intrinsic to LLMs. ULTRA first sequentially reads text chunks of a document to generate a candidate argument set, upon which ULTRA learns to drop non-pertinent candidates through self-refinement. We further introduce LEAFER to address the challenge LLMs face in locating the exact boundary of an argument span. ULTRA outperforms strong baselines, which include strong supervised models and ChatGPT, by 9.8% when evaluated by the exact match (EM) metric.