I'm currently working on my game for the js13k competition. As you might infer from the compo's name, you only have 13k to make your game, which rules out most frameworks. I've done this a few times before, and because of the restrictions, the things I end up spending time on are a bit different from more "standard" jams. For example:
Yesterday, I looked at how to do graphics. I want to have some minimal pixel art in my game, but even small images can take up significant space, so I want a solution as cheap as possible in terms of space. I looked at creating an Image object dynamically, which lead me to looking at the data structure and the getImageData() and putImageData() methods - but I concluded that creating my own structure and just drawing the pixels individually is a cheap and good enough way for what I need.
Before that, I spent a fair amount of time researching how fonts work in HTML5 Canvas and especially how to not anti-alias them and how to make them crisp and clear even with a small resolution. There are some libraries that will give you a cross-browser/OS consistent font, but I don't want to dedicate too much space to this.
Today I am looking at mouse input. If I didn't use canvas, it would be easy, I could just attach events on images or buttons. However, in canvas you basically only get a single event (there is a addHitRegion() API but it's currently only available in experimental builds of Chrome and Firefox). Looks like I'll have to write my own thing to temporarily add a thing you can click on.