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Traditional query processing relies on engines that are carefully optimized and engineered by many experts. However, new techniques and user requirements evolve rapidly, and existing systems often cannot keep pace. At the same time, these systems are difficult to extend due to their internal complexity, and developing new systems requires substantial engineering effort and cost. In this paper, we argue that recent advances in Large Language Models (LLMs) are starting to shape the next generation of query processing systems. We propose using LLMs to synthesize execution code for each incoming query, instead of continuously building, extending, and maintaining complex query processing engines. As a proof of concept, we present GenDB, an LLM-powered agentic system that generates instance-optimized and customized query execution code tailored to specific data, workloads, and hardware resources. We implemented an early prototype of GenDB that uses Claude Code Agent as the underlying component in the multi-agent system, and we evaluate it on OLAP workloads. We use queries from the well-known TPC-H benchmark and also construct a new benchmark designed to reduce potential data leakage from LLM training data. We compare GenDB with state-of-the-art query engines, including DuckDB, Umbra, MonetDB, ClickHouse, and PostgreSQL. GenDB achieves significantly better performance than these systems. Finally, we discuss the current limitations of GenDB and outline future extensions and related research challenges.