Forgot to post this when I did it, but I had fun trying out this contest without touching the provided Sentinel Imagery. Instead, I used Google Earth Engine to access some landsat images and sample them without needing to download any big files. There is no reason this couldn't be done with Sentinel Imagery (also available in GEE) but this was mainly just a proof of concept. I wrote up my experiments in two quick blog posts:

Initial post: https://datasciencecastnet.home.blog/2019/06/26/tutorial-predicting-crop-types-with-gee/

Follow-up with better modelling and submissions: https://datasciencecastnet.home.blog/2019/06/27/tutorial-improving-crop-type-predictions/

This approach scores ~1.41 despite only using a few low-quality 30m resolution images. A fun PoC of what's doable without much effort (and with terrible internet). I hope someone finds it interesting!