Is it possible to use Drupal with CouchDB?
There are a few things to keep in mind.

Even though Drupal support multiple databases (mainly MySQL,  PostgreSQL, and now anything supported by PDO), this is only true of the core files.  There are contributed modules that use database specific syntax and/or functions.  In addition, there are some functions that show subtle differences when run on different database servers (for example, see Does db_affected_rows() works well with PostgreSQL?).

The modules for non-SQL databases are primarily for offloading some core tables from the primary database. You still need an SQL database when using these.

Personally, I don't mess around with any of these other databases. I may start experimenting with MongoDB, but I can't really use it with client's projects as its support on commodity hosting is limited. Setting it up for them would eat into their budgets, and clients would rather have me do Drupal work than IT work.