As the title says, how long can I keep pureed root ginger in the fridge?
If you have a lot or don't anticipate using it in the next 5 days, then freeze it. If it's already minced or pureed then simply measure it into quantities that will be the easiest for you to use and then freeze them on a tray lined with plastic wrap. Once frozen, remove and put into a self-sealing bag and keep in the freezer for future use.
Whole ginger can be frozen simply as it is. Just put it a bag in the freezer. Galangal for example, is commonly sold frozen in asian markets You can pull it out and grate what you need on a rasp style zester while frozen. I never even peel ginger as most uses for it require it to either be minced finely or strained out all together. A little gingerroot skin is just more fiber for the diet.
I don't think preservation is the issue you want to worry about, it is taste.
As with most fresh vegetable matter, you should have no trouble telling if that ginger has become unwholesome. It's not like canned goods, where you have to worry about botulism. If it's bad, it will look and smell bad.
Long before it's bad, however, it will be off.
Personally, I only grate what I need. You can always grate a bit more if you're low.
If for some reason you are sitting on 250g of grated ginger due to an overenthusiastic assistant or something, I'd personally try to turn it into gingerbread within about 2 days.