I apologize for going against your bold text, since I don't have any first-hand experience, but this set of blog posts is a good example of solving a problem with CouchDB.
CouchDB: A Case Study
Essentially, the textme application used CouchDB to deal with their exploding data problem. They found that SQL was too slow to deal with large amounts of archival data, and moved it over to CouchDB. It's an excellent read, and he discusses the entire process of figuring out what problems CouchDB could solve and how they ended up solving them.
