I generally go about 5-10 games/skates between sharpens, but sometimes longer when I get lazy/cheap. I have gone 20+ games before, I definitely get less bite but it's still manageable, unless you get actual nicks in the blade of course.
And yeah, blueberrydanish, if you find skates too catchy/edgy after a sharpen, take the advice of others in this thread and try a 5/8" ROH (and if that's still too much bite, try 3/4"). Also, make sure to find a good sharpener and stick with them, a bad sharpen is gonna feel terrible regardless of the ROH, and a good sharpen should feel pretty smooth even with a 3/8" or 1/2" ROH (though it's all preference).