How long can I keep pureed root ginger
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.