As little as possible, lol. But I do keep some. I tear out blank pages from work-books and add them to the scratch paper pile, that reduces it a lot. Anything I don't like, I bin (when they're not looking!!). Some of it (stuff I especially like) goes up on their walls (replacing older stuff). What I can, I pile neatly in their rooms, with previous yrs things. Occasionally I go thru the piles and cull stuff that doesn't seem so exciting any more. Something like reading records I always keep because it's fun to compare at what age/point different DC read the same book(s).
We are lucky - the children at the DDs school make what they call their special books in the last week of the school year they are large scrapbooks with fabric covers made by the children (usually batik or something) and all their best work from the year pasted in. I know people who were at the school 15/20 years ago who still have all their 'special books'.
We were finding the same thing, with MASSES of work coming home, and since this country seems so set on sending things home at the end of the year (which I will never understand the reason for this, but to each their own ) so for the past two years, I told DS and DD that they could pick a max of 5 favourite pieces of work from each subject area, and we made a "scrapbook" over the course of the summer. It has become a tradition they both look forward to, and it gives them the opportunity to go through their work again and we discuss what they liked or did not like about different things.