<div id="a">This is some
   <div id="b">text</div>
</div>

Getting "This is some" is non-trivial. For instance, this returns "This is some text":

driver.find_element_by_id('a').text

How does one, in a general way, get the text of a specific element without including the text of it's children?

(I'm providing an answer below but will leave the question open in case someone can come up with a less hideous solution).

Here's a general solution:

def get_text_excluding_children(driver, element):
    return driver.execute_script("""
    return jQuery(arguments[0]).contents().filter(function() {
        return this.nodeType == Node.TEXT_NODE;
    }).text();
    """, element)

The element passed to the function can be something obtained from the find_element...() methods (i.e. it can be a WebElement object).

Or if you don't have jQuery or don't want to use it you can replace the body of the function above above with this:

return self.driver.execute_script("""
var parent = arguments[0];
var child = parent.firstChild;
var ret = "";
while(child) {
    if (child.nodeType === Node.TEXT_NODE)
        ret += child.textContent;
    child = child.nextSibling;
}
return ret;
""", element) 

I'm actually using this code in a test suite.