geom_smooth() what are the methods available?

The method argument specifies the parameter of the smooth statistic. You can see stat_smooth for the list of all possible arguments to the method argument.

The se argument from the example also isn't in the help or online documentation.

When 'se' in geom_smooth is set 'FALSE', the error shading region is not visible

Sometimes it's asking the question that makes the answer jump out. The methods and extra arguments are listed on the ggplot2 wiki stat_smooth page.

Which is alluded to on the geom_smooth() page with:

"See stat_smooth for examples of using built in model fitting if you need some more flexible, this example shows you how to plot the fits from any model of your choosing".

It's not the first time I've seen arguments in examples for ggplot graphs that aren't specifically in the function. It does make it tough to work out the scope of each function, or maybe I am yet to stumble upon a magic explicit list that says what will and will not work within each function.