PostgreSQL
PostgreSQL is an open-source relational database that handles transactional workloads with ACID guarantees, foreign keys, MVCC concurrency control, and a large extension ecosystem. Beyond plain tables it supports JSONB documents, full-text search, range and geometric types, window functions, and materialized views, so many specialized workloads fit without a second database.
It is a common choice for web applications, content systems, reporting, and geospatial data, with a standards-oriented SQL dialect and extensive documentation.
Hosting it yourself keeps records and backups on your own storage, adds no per-connection fees, and leaves dump and physical backup tools free and straightforward to automate.