I've memorized a few small API's ... I thought it was normal ...
For small API's it is because there's just not much to pick up, but it would be a waste of time to intentionally memorize every API call in OpenGL or something. Knowing what's available in the API is more important than being able to make obscure calls without referring to the docs. Especially with IDE features like intellisense (or equivalents) being able to fill in the blanks with a lot of stuff.
Basically you don't want to memorize a bunch of stuff that's going to be outdated when the next version rolls around. You've got better things to do. ;)
BTW - for the sake of the meme - "One does not simply.... program a game without a database"
That's a silly quote. Are you saying an Angry Birds clone needs a database?
Or are you trying to say a database is simply external data that drives the game content; in that case, I'd agree, but in the case of traditional databases, you can make tons of fun games without them.
For real, I don't know why it seems like everyone is obsessed with formal databases these days. There are certainly uses for them but it's not as if languages don't have other highly effective methods for storing and retrieving data.