The optimal number of connections varies based on the DB engine. For example, Postgres recommends the following:
A formula which has held up pretty well across a lot of benchmarks for years is that for optimal throughput the number of active connections should be somewhere near ((core_count * 2) + effective_spindle_count). Core count should not include HT threads, even if hyperthreading is enabled. Effective spindle count is zero if the active data set is fully cached, and approaches the actual number of spindles as the cache hit rate falls. Benchmarks of WIP for version 9.2 suggest that this formula will need adjustment on that release. There hasn’t been any analysis so far regarding how well the formula works with SSDs.
You can find detailed explanations here: Number Of Database Connections - PostgreSQL wiki
At the end it is like Jose mentioned start with 10 and experiment (although 10 is rather arbitrary number but the formula above is a better starting point). Also consider that you might be running multiple instances of the service that talk to one DB - the formula described above is per DB instance not per clients instances it serves. So, there is plenty to experiment with.
