I have saved everything I've ever written.
Really old first drafts that are absolutely horrible, little beginnings of stories, random bits that start in the middle and go 'til the end, illustration-laden first attempts at books, terrible poems, and every once in a while, a jewel that I'll look back on and read and wish I could use.
This technique has two names: the first is "hoarding," but the term I prefer is (to quote Roy Peter Clark) is "saving string."
Why save that junk, you wonder? Well, because it's very sentimental, or very good, or useful.
There are lots of reasons to save string, but two prevail in my mind: the first is because an immense case of writer's block stopped you dead in your tracks and you're hoping that someday when you're a famous author you can come back to "the idea that got away" and give it a second chance, because by then you'll at least know what the heck you're doing. The second is because it's really good but very personal so you can never show anyone lest you be forced to explain what type of insanity induced you to write such a thing. (Of course, the things you write for others are never as good as the things you write for yourself.)
You might be super messy and think, "I never want to look at my old drafts again, much less try to find a place to keep them! They'll get lost in a week!" Well, maybe you don't want to look at them now, but someday you'll have that zap of inspiration and be moved to finish one of those pieces of junk. And it is true things get lost, I have a few excellent manuscripts that fell behind my bed and now I'll never see them again. Here's 3 tips: Binders. Clips. Staples. Those are harder to lose.
You also might be a total neat freak and say, "I never want to look at my old drafts again, much less cover my desk in them!" Same advice as above. My desk is only mess when I'm caught in the act of writing - everything has a place otherwise.
It's not that hard to save string, and it's actually very useful. Take one of my characters, for example: once I wrote about half a book, and one of the main characters was a black-haired, blue-eyed girl with water-powers. Her name was Eva. Well, I eventually gave up on that book (which I thought would be my masterpiece, despite a bad case of plagiarism), and I scrapped the characters. Well, eventually I took my illustration of Eva and matched her looks and powers to my new character's personality, and they clicked.
Take a peek at your old writing and see what you've got. What characters, settings, plots really shine? What can you reuse? What, looking back on your writing, worked? What were you really good at? Maybe that ream of fantasy destined for the trash heap is really your next big idea in disguise.