Not all databases systems require a connection pool, though it is often a good idea.

I disagree. I've faced incidents on oracle/sql server/postgres/sybase instances when several hundreds of connections where spawned.

You definitely need to think about pooling fro the beginning of the design of your application.

Yes, all of those databases allocate query memory for queries at connection time, and consequently don't scale efficiently for large number of idle clients. That's not all databases. Some don't do that. Some don't even do connections period.

Connection pools are a necessary work around for a specific design decision with the system software. They can be helpful other problems as well, but they aren't necessarily a requirement.