Real-world financial filings report critical information about an entity's investment holdings, essential for assessing that entity's risk, profitability, and relationship profile. Yet, these details are often buried in messy, multi-page, fragmented tables that are difficult to parse, hindering downstream QA and data normalization. Specifically, 99.4% of the tables in our financial table dataset lack bounding boxes, with the largest table spanning 44 pages. To address this, we present TASER (Table Agents for Schema-guided Extraction and Recommendation), a continuously learning, agentic table extraction system that converts highly unstructured, multi-page, heterogeneous tables into normalized, schema-conforming outputs. Guided by an initial portfolio schema, TASER executes table detection, classification, extraction, and recommendations in a single pipeline. Our Recommender Agent reviews unmatched outputs and proposes schema revisions, enabling TASER to outperform vision-based table detection models such as Table Transformer by 10.1%. Within this continuous learning process, larger batch sizes yield a 104.3% increase in useful schema recommendations and a 9.8% increase in total extractions. To train TASER, we manually labeled 22,584 pages and 3,213 tables covering $731.7 billion in holdings, culminating in TASERTab to facilitate research on real-world financial tables and structured outputs. Our results highlight the promise of continuously learning agents for robust extractions from complex tabular data.