What I tell a bored beginner is: learn to knit on double-pointed needles!
Then suddenly, a whole wealth of "intermediate" patterns opens up to you! There are plenty of patterns in the free patterns pages that are called "intermediate" only because they use DPN's. Otherwise, they're very basic patterns!
Hats are the perfect example. Most hats are super easy. It's just all knitting, plus a decrease used repeatedly to taper in the top. And they're nice and quick! Much smaller than a scarf!
Mittens are not much harder, if you have some simple directions. It's just knitting around and around on DPNs, first doing the cuff, then increasing in two places repeatedly to make the base of the thumb. Then putting those thumb stiches you've created on hold on a string, while you knit around and around the rest of the hand, and decrease to taper the top. Then you come back to those thumbs stitches and knit them, around and around, until you're at the top of the thumb, and then decrease that. It looks complicated, but it's really not hard to do. And they're impressive as gifts!
Hope that helps!
Happy Knitting and Happy Holidays!