Personally, I have a soft spot for CODASYL databases, where you walk datasets to access related data elements... perhaps because my first experience with databases was DECs DBMS. The ability to locate a parent record, then simply walk a set of children was simple to use; and the explicit relationships between datasets built into the underlying database structure is something that is sadly lacking in relational databases... and before anybody suggests it, foreign keys are a pale, one-dimensional shadow of that relationship.
The complexities of inner/outer/straight joins in SQL become an irrelevance, because the structure of the dataset maintains the relationships between different record sets (tables), allowing you to access only those dataset records that are related. Lookup an order and Walking the OrderLine dataset would return only the Order Lines that were part of that order, with no need to formulate a second query to retrieve that series of rows from the entire table of order lines for all orders.
The drawback was that almost all the design needed to be done up front... once the database structure was designed it was pretty much fixed, hard to change.... compared with RDBMS where it's much easier to add new tables at any time.... not particularly suited to a world of RAD/Agile/XP/Scrum.
Now if only somebody could come up with a database system where the structure was as flexible and easy to change as an RDBMS, with all the simplicity of data access of a CODASYL database.
OLAP databases are another great alternative to RDBMS, where the data can readily be structured as an n-dimensional cube, where a query allows you to "slice and dice" the cube, quickly extracting segments of the data. My first introduction to OLAP was Oracle Express, which (unfortunately) uses its own proprietary query language rather than the de facto standard MDX. However, not all data can readily by fitted to a "cube" structure, so it's only suitable for certain application types - although almost any data mining applications are well suited.