How long does it take before the "newness" of a relationship wears off?

If you are lucky and if he/she is the right one.. then the newness just transforms into something deeper, more close and more wonderful... The excitement need not go away and sometimes newness can mean a slight awkwardness so to be honest, getting past that stage can be a fabulous thing :)

The first time they ask you for money!

The moment you let it.

That all depends on the persons in the relationship. Some wear off faster than others. 
Sometimes, the the newness stays there forever because you love them so much that everyday becomes an adventure.

In some cases Princess...never.

It's relative.  There is no real answer.  Probably the best possible answer is Carbon's.

When you either get comfortable with them or they start to annoy you.

3rd date

not yet